mathematics syllabus - dr. b.r. ambedkar university

1

DEPARTMENT OF MATHEMATICS

Dr. B. R. AMBEDKAR UNIVERSITY – SRIKAKULAM

Outcome Based Curriculum

Mathematics Programme

(With Effect From 2019-20 Admitted Batch)

Dr. B.R. Ambedkar University, Srikakulam

Etcherla– 532410

2

INDEX

S.No Content Page

1 About Department 5

2 University Vision, Mission

and Department Vision, Mission

6

3 SWOT/C Analysis 7

4 Strategic Plans 8

5 Short and Long Term Goals 9

6 Programme Educational Objectives and Programme

Outcomes 10

7 General Regulations 12-15

8 First Year – First semester credit system 16

9 First Year – Second semester credit system 17

10 Second Year – Third semester credit system 18

11 Second Year – Fourth semester credit system 19

12 M101 Algebra - I 20-21

13 M102 Real Analysis – I 22-23

14 M103 Topology 24-25

15 M104 Differential Equations 26-27

16 M105 Linear Algebra 28-29

17 M201 Algebra – II 30-31

18 M202 Real Analysis – II 32-33

19 M203 Probability 34-35

20 M204 Complex Analysis 36-37

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21 M205 Graph Theory 38-39

22 M301 Functional Analysis 40-41

23 M302 Number Theory 42-43

24 M303 Classical Mechanics 44-45

25 M304(1) Operations Research 46-47

26 M304(2) Mathematical Biology 48-49

27 M304(3) Commutative Algebra - I 50-51

28 M304(4) Banach Algebras 52-53

29 M305(1) Numerical Analysis 54-55

30 M305(2) Mathematical Software 56-57

31 M305(3) Fuzzy Set Theory 58-59

32 M305(4) Universal Algebra 60-61

33 M401 Measure and Integration 62-63

34 M402 Partial Differential Equations 64-65

35 M403 Mathematical Methods 66-67

36 M404(1) Lattice Theory 68-69

37 M404(2) Theory of Computations 70-71

38 M404(3) Commutative Algebra-II (Prerequisite

Commutative Algebra – I)

72-73

39 M404(4) Theory of Linear Operators 74-75

40 M405(1) Wavelet Analysis 76-77

41 M405(2) Programming in C 78-79

42 M405(3) Semi Groups 80-81

4

43 M405(4) Financial Mathematics 82-83

44 Model Papers

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ABOUT DEPARTMENT

Department of Mathematics was established in the academic year 1988-1989 and it offers

opportunities for the education and research in a wide spectrum of areas in Mathematics and

Applied Mathematics. The Department started offering CBCS elective courses for

postgraduate programs and also advanced courses for Ph. D. program. The Department of

Mathematics strives to be recognized for excellence among academic institutions in India.

The Department wishes to focus on providing a comprehensive curriculum at postgraduate

levels and career opportunities in India. The department is committed to train the students to

make them motivated and dedicated teaching and scientific research.

The department is offering two programmes one is M.ScMathematics and other one is

M.ScApplied Mathematicsandhas been introduced choice based credit system curriculum.

The new syllabus that is proposed for the M.Sc Mathematics and M.Sc Applied Mathematics

curriculum has been prepared keeping in view of the guidelines presented in UGC model

curriculum for PG programme. The proposed syllabus covers 95% of the content suggested

by UGC. This also covers the syllabus that is needed to clear the examinations conducted at

national level such as CSIR-NET/SET examination for research fellowships, eligibility for

lecturer-ship, GATE examination for higher studies in technical education, also employed in

various government and non government organizations.

This syllabus is oriented to create suitable workforce to support teaching community in

Mathematics as well as in Applied Mathematics. The courses involved in the curriculum have

tremendous potential for applications in the industry. Some of these courses are useful in

designing algorithms and performing computations needed in several real world applications.

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Vision of the Department

To make the department as a global centre

for excellence in Mathematics to

coordinate the growth of Science and

Technology.

University Mission

Mitigating the economic and social

sufferings of the region by invoking the

strengths of faculty through community

oriented actions by optimal usage of

human resources.

University Vision

Creation of an enabling environment

where in universities would act as agents

of social change and transformation

through innovativeness and outreaching

and make it a “People’s University”.

Mission of the Department

To impart quality education and scientific

researchin Mathematics through updated

curriculum, effective teaching learning

process. To inculcate innovative skills,

theme of team-work and ethical practices

among students so as to meet societal

expectations.

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SWOT/C Analysis

Strengths

1. The major strength of the Department of Mathematics is the quality of the faculty. A

majority of the faculty have strong and established research programs with strengths in the

areas of nonlinear analysis, applied analysis, mathematical modelling and simulation,

Mathematical Biology, Lattice Theory.

2. Quality teaching has been a priority for mathematics faculty, and several faculties have

received outstanding research awards.

3. Strong faculty qualification, committed, talented and dedicated, knowledgeable and

cooperative faculty, great collaboration and good communication among faculty.

4. Faculty-initiated activitiesoutside class like weeklyseminar and studentresearch.

Weaknesses

1. Difficult to communicate to higher authorities in proper way.

2. Failed to eliminate chalkboards for teaching.

3. Lack of regular faculty.

Opportunities

1. Economy encouraging more students to choose Ship, raising enrolment.

2. Current trends in improvingSTEM education provide grantopportunities.

3. Promote more than math minimum

4. Competitive enrolment (better prerequisite enforcement)

5. Better collaboration with other disciplines.

Threats

1. Student attitude towards mathematics

2. Lack of student dedication

3. Lack of technology (hardware /software)

4. Lack of computer lab & library in the department

5.Administration fails to involve proper individuals before making decisions

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Strategic Plans

The Faculty of Mathematics Strategic Plan 2019-2024 was developed through extensive

consultation with Mathematics faculty, students, staff, university administrators, national

researchers, employers and alumni. The purpose is to formulate objectives for the

department's future research profile and to bring new study programs and new educational

approaches into the department's educational profile.

1. Fill open and anticipated positions with faculty who excel in both research and teaching.

2. Continue to make connections with industries, labs, and government agencies that can

offer our students internships and regular employment. Bring speakers and recruiters to

campus when possible.

3. Introduce more innovative teaching methods to some of our coursework (blended

[online/classroom] learning, the Moore method, etc.) to encourage more intellectual

engagement.

4. Continue to emphasize the importance of external funding.

5. Be one of the top research departments in the world, and be widely recognized as such.

6. Develops world class facilities for collaborative research and learning in mathematics.

7. Maintain an existing, dynamic, collaborative environment supporting activities at he

frontiers of the mathematical sciences.

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Short Term Goals

1. To get regular faculty

2. To start a new post graduates programme in Mathematics.

3. To build on inter disciplinary programmes with other departments within the University.

4. Prepare graduate students for leadership in both academic and non-academic career paths

in an increasingly interdisciplinary world.

Long term Goals

1. To attract and retain academics of a high calibre on the Department’s faculty.

2. To attract motivated and talented students to the master’s and doctoral programmes of the

Department.

3. To provide the best possible facilities for our faculty and students, particularly in the areas

of computer facilities, library facilities and administrative support.

4. To create an environment that supports outstanding research.

5. To pursue collaborative programmes with highly reputed national institutions.

6. To provide a simulating teaching environment for the post graduate students of the

department.

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M.Sc Mathematics Programme Educational Objectives

The objectives of the M.Sc. Mathematics program are to develop students with the following

capabilities

1. To produce PG students with high integrity and good ethics and to train students to deal

with the problems faced by software industry through knowledge of mathematics and

scientific computational techniques.

2. The Department wishes to focus on providing a comprehensive curriculum at postgraduate

levels and career opportunities in India.

3. The graduates will work and communicate effectively in intra-disciplinary,

interdisciplinary environment, either independently or in team, and demonstrate leadership

quality in area of Mathematics.

M.Sc Mathematics Programme Outcomes

The successful completion of this program will enable the students to

1. Apply a wide range of mathematical techniques and application of mathematical

methods/tools in other scientific and engineering domains.

2. Gain the knowledge of contemporary issues in the field of Mathematics and applied

sciences.

3. Understand the scientific theories and methods, gain experience in working independently

with scientific questions and engineering problems, and clearly express their opinion on

academic issues.

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Dr. B. R. AMBEDKAR UNIVERSITY, SRIKAKULAM

General Regulations relating to

POST GRAUDATE AND PROFESSIONAL COURSES

Syllabus under Credit Based Semester System

(With effect from 2019-2020 admitted batch)

1. Candidates seeking admission for the Masters/Professional Degree Courses shall be required to have passed the qualifying examination prescribed for the course of any University recognized by Dr. B.R. Ambedkar University, Srikakulam as equivalent there to.

2. The course and scope shall be as defined in the Scheme of Instruction and syllabus

prescribed.

3. The course consists of 2/4/6 semesters, @ two semesters/year, unless otherwise

specified.

4. The candidates shall be required to take an examination at the end of each semester of the study as detailed in the Scheme of Examination.

i. (a). Each semester theorypaper in M.Sc Mathematics/M.Sc Applied

Mathematics programme except M305(2) Mathematical Software (3rd semester) and M405(2) Programming in C (4th semester) carries a maximum of 100 marks, of which 75 marks shall be for semester-end theory examination of the paper of three hours duration and 25 marks shall be for internal assessment.

(b). M305(2) Mathematical Software (3rd semester) and M405(2) Programming in C (4th semester) carries a maximum of 100 marks, of which 50 marks shall be for semester-end theory examination paper of the of 3hrs duration, 25 marks shall be for internal assessment and 25 marks of which 5 marks shall be for Lab, 5 marks shall be for observation, 5 marks shall be for record work and 10 marks shall be for viva examination (External).

ii. Internal Assessment for 25 Marks: Three mid-term exams, two conventional

(descriptive) for 15 marks and the third – ‘on-line’ with multiple choice questions for 5 marks for each theory paper shall be conducted and 5 marks for student Assignment submission for each course. The average of these first two mid-term and the marks in the online mid exams shall be taken as marks obtained for the paper under internal assessment. If any candidate appears for only one mid-term exam, the average mark, dividing by two shall be awarded. If any candidate fails to appear for all the midterm exams of a paper, only marks obtained in the theory paper shall be taken into consideration for declaring the result. Each mid-term exam shall be conducted only once.

iii. Candidates shall be declared to have passed each theory paper if he/she

obtains not less than E Grade ie., an aggregate of 40 % of the total marks inclusive of semester-end and internal assessment marks in each paper.

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5. A candidate appearing for the whole examination shall be declared to have passed the examination if he/she obtains a Semester Grade Point (SGP) of 5.0 and a CGPA of 5.0 to be declared to have passed the Course.

6. Notwithstanding anything contained in the regulations, in the case of Project

Report/Dissertation/ Practical/Field Work/Viva-voce etc., candidates shall obtain not less than D grade, i.e., 50% of marks to be declared to have passed the examination.

7. ATTENDANCE: Candidates shall put in attendance of not less than 75% of attendance, out of the total number of working periods in each semester. Only such candidates shall be allowed to appear for the semester-end examination.

(a) A candidate with attendance between 74.99% and 66.66% shall be allowed to appear for the semester-end examination and continue the next semester only on medical and other valid grounds, after paying the required condonation fee.

(b) In case of candidates who are continuously absent for 10 days without prior permission on valid grounds, his/her name shall automatically be removed from the rolls.

(c) If a candidate represents the University at games, sports or other officially organized extra-curricular activities, it will be deemed that he/she has attended the college on the days/periods

8 Candidates who put in a minimum of 50% attendance shall also be permitted to continue for the next semester. However, such candidates have to re-study the semester course only after completion of the course period for which they are admitted. The candidate shall have to meet the course fees and other expenditure.

9 Candidates who have completed a semester course and have fulfilled the necessary

attendance requirement shall be permitted to continue the next semester course irrespective of whether they have appeared or not at the semester-end examination, at their own cost.

Such candidates may be permitted to appear for the particular semester-end

examination only in the following academic year; they should reregister/ reapply for the Semester examination.

The above procedure shall be followed for all the semesters

10. Candidates who appear and pass the examination in all the papers of each and every

semester at first appearance only are eligible for the award of Medals/Prizes/Rank Certificates

11. BETTERMENT: Candidates declared to have passed the whole examination may reappear for the same examination to improve their SGPA, with the existing regulations without further attendance, paying examination and other fees. Such reappearance shall be permitted only with in 3 consecutive years from the date of first passing the final examination. Candidates who wish to appear thereafter should take the whole examination under the regulations then in vogue.

12. The semester-end examination shall be based on the question paper set by an external paper-setter and there shall be double valuation for post-Graduate courses. The concerned Department has to submit a panel of paper-setters and examiners approved by the BOS and the Vice-chancellor nominates the paper-setters and examiners from the panel.

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13. In order to be eligible to be appointed as an internal examiner for the semester-end

examination, a teacher shall have to put in at least three years of service. Relaxation of service can be exempted by the Vice-Chancellor in specific cases.

14. If the disparity between the marks awarded in the semester-end examination by

internal and external examiners is 25% or less, the average marks shall be taken as the mark obtained in thepaper. If the disparity happens to be more, the paper shall be referred to another examiner for third valuation. In cases of third valuation, of the marks obtained either in the first or second valuation marks, whichever is nearest to the third valuation marks are added for arriving at the average marks.

15. Candidates can seek revaluation of the scripts of the theory papers by paying the

prescribed fee as per the rules and regulations in vogue.

16. The Project Report/Dissertation/ Practical/Field Work/Viva-voce etc shall have

double valuation by internal and external examiners.

17. A Committee comprising of the HOD, one internal teacher by nomination on rotation

and one external member, shall conduct viva-voce examination. The department has to submit the panel, and the Vice-chancellor nominates viva-voce Committee.

18. Grades and Grade Point Details (with effect from 2019-20 admitted batches)

S.No. Range of Marks% Grade Grade Points

1. > 90 ≤100 O 10.0 Out Standing

2. > 80 ≤ 90 A+ 9.0 Excellent

3. > 70 ≤80 A 8.0 Very Good

4. > 60 ≤70 B+ 7.0 Good

5. > 55 ≤60 B 6.0 Above Average

6. > 50 ≤55 C 5.0 Average

7. ≥ 40 ˂50 D 4.0 Pass

8. ˂40 F 0.0 Fail

9. 0.0 Absent

19. Calculation of SGPA (Semester Grade Point Average) & CGPA (Cumulative Grade

Point Average):

For example, if a student gets the grades in one semester A,A,B,B,B,D in six subjects having credits 2(S1), 4(S2), 4(S3), 4(S4), 4(S5), 2(S6), respectively. The SGPA is calculated as follows:

{ 9(A)x2(S1)+9(A)x4(S2)+8(B)x4(S3)+8(B)x4(S4)+8(B)x4(S5)+6(D)x2(S6)} 162

SGPA = --------------------------------------------------------------------------- = ------ = 8.10

{2(S1) +4(S2) +4(S3) +4(S4) +4(S5) +2(S6)} 20

i. A student securing ‘F’ grade thereby securing 0.0 grade points has to appear and

secure at least ‘E’ grade at the subsequent examination(s) in that subject.

ii. If a student gets the grades in another semester D, A, B, C, A, E, A, in seven subjects

having credits 4(S1),

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2(S2), 4(S3), 2(S4), 4(S5), 4(S6), 2(S7) respectively,

{6(D)x4(S1)+9(A)x2(S2)+8(B)x4(S3)+7(C)x2(S4)+9(A)x4(S5)+5(E)x4(S6)+9(A)x2(S7)} 162

SGPA = --------------------------------------------------------------------------------------------------------- = ------ = 7.36

{4(S1) +2(S2) +4(S3) +2(S4) +4(S5) +4(S6) +2(S7)} 22

(9x2+9x4+8x4+8x4+6x2+6x4+9x2+8x4+7x2+9x4+5x4+9x2) 324

CGPA = ------------------------------------------------------------------------------ = -------- = 7.71

(20+22) 42

a) A candidate has to secure a minimum of 5.0 SGPA for a pass in each semester in case

of all PG and Professional Courses. Further, a candidate will be permitted to choose any paper(s) to appear for improvement in case the candidate fails to secure the minimum prescribed SGPA/CGPA to enable the candidate to pass at the end of any semester examination.

b) There will be no indication of pass/fail in the marks statement against each individual

paper.

c) A candidate will be declared to have passed if a candidate secures 5.0 CGPA for all

PG and Professional Courses.

d) The Classification of successful candidates is based on CGPA as follows:

i) Distinction –CGPA 8.0 or more;

ii) First Class –CGPA 6.5 or more but less than 8.0

iii) Second Class –CGPA 5.5 or more but less than 6.5

iv) Pass –CGPA 5.0 or more but less than 5.5

e) Improving CGPA for betterment of class will be continued as per the rules in vogue.

f) CGPA will be calculated from II Semester onwards up to the final semester. CGPA

multiplied by gives“10” aggregate percentage of marks obtained by a candidate.

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DEPARTMENT OF MATHEMATICS

Dr. B. R. AMBEDKAR UNIVERSITY – SRIKAKULAM CURRICULUM STRUCTURE FOR CHOICE BASED CREDIT SYSTEM

(CBCS)

(W.E.F. 2019-20 ADMITTED BATCH)

.

M.Sc Mathematics FIRST YEAR – FIRST SEMESTER

Paper

Code

Title of the paper

Core papers

Credit

value

Lectures

per

week

Max

Marks

External

Assessment

Internal

Assessment

Lab

Assess

ment

M 101 Algebra – I 4 6 100 75 25 0

M 102 Real Analysis – I 4 6 100 75 25 0

M 103 Topology 4 6 100 75 25 0

M 104 Differential

Equations 4 6 100 75 25 0

M 105 Linear Algebra 4 6 100 75 25 0

SD 106 Communication

Skills - I 2 3 100 75 25 0

FW-1 Extension Activity I 2 4 25 0 25 0

Internship 1 0 - - - -

Total 25 - 625 - - -

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M.Sc Mathematics FIRST YEAR – SECOND SEMESTER

Paper

Code

Title of the

paper

Core papers

Credit

value

Lectures

per

week

Max

Marks

External

Assessment

Internal

Assessment

Lab

Assessment

M 201 Algebra – II 4 6 100 75 25 0

M 202 Real Analysis –

II 4 6 100 75 25 0

M 203 Probability 4 6 100 75 25 0

M 204 Complex

Analysis 4 6 100 75 25 0

M 205 Discrete

Mathematics 4 6 100 75 25 0


Skills - II 2 3 100 75 25 0

FW-2 Extension

Activity II 2 4 25 0 25 0

MOOCS MOOCS 2 0 - - - -

Internship 1 0 - - - -

Total 27 - 625 - - -

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M.Sc Mathematics SECOND YEAR – THIRD SEMESTER

Paper

Code

Title of the paper

Core papers

Credit

value

Lectures

per week

Max

Marks

External

Assessment

Internal

Assessment

Lab

assessment

M 301 Functional Analysis 4 6 100 75 25 0 M 302 Number Theory 4 6 100 75 25 0 M 303 Classical Mechanics

4 6 100

75 25 0

Elective Papers

(Stream

A)

M 304

M304(1) Operations

Research

4

6

100

75

25

0

M304(2) Mathematical

Biology

M304(3) Commutative

Algebra-I

M304(4) Banach

Algebras

(Stream

B)

M 305

M305(1)Numerical

Analysis

4

6

100

75

25

0 M305(2)Mathematical

Software

50

25

25 M305(3) Fuzzy Set

Theory

75

25

0 M305(4) Universal

Algebra

75

25

0 SD 306 Communication

Skills - III 2 3 100 75 25 0

FW-3 Extension Activity III 2 4 25 0 25 0 MOOCS MOOCS 2 0 - - - -

Internship 1 0 - - - - Total 27 - 625 - - -

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M.Sc Mathematics SECOND YEAR – FORTH SEMESTER

Paper

Code

Title of the paper

Core papers

Credit

value

Lectures

per week

Max

Marks

External

Assessment

Internal

Assessment

Lab

Assessment

M 401 Measure and Integration 4 6 100 75 25 0 M 402 Partial Differential

Equations 4 6 100 75 25 0

M403 Mathematical Methods 4 6 100 75 25 0 Elective Papers

(Stream

A)

M 404

M404(1) Lattice Theory

4

6

100

75

25

0

M404(2) Commutative

Algebra-II (Prerequisite

Commutative Algebra –

I)

M404(3) Theory Of

Computations

M404(4) Theory of

Linear Operators

(StreamB)

M 405

M405(1) Wavelet

Analysis

4

6

100

75

25

0 M405(2)Programming in

C 50 25 25

M405(3)Semi Groups

75

25

0 M405(4)Financial

Mathematics 75 25 0


Skills - IV

2 3 100 75 25 0

FW-4 Extension Activity IV 2 4 25 0 25 0 MOOCS MOOCS 2 0 - - - -

Internship 1 0 - - - - Viva-voce 4 0 100 100 - - Total 31 - 725 - - -

Total Marks: 2600

Total Credits: 110

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M.Sc Mathematics FIRST YEAR – FIRST SEMESTER

Course Code

&Course Name M 101

Algebra - I

Objectives

1. The primary goal in Algebra - I is to help students transfer their concrete mathematical

knowledge to more abstract algebraic generalizations.

2. Algebra - I is designed to give students a foundation for all future mathematics courses.

3. Students will explore: Basic set theory. Groups- Some Examples of Groups- Some preliminary

Lemmas- Subgroups-A counting principle- Normal subgroups and Quotient Groups,

homomorphism, Isomorphism, automorphism and its applications.

4. Throughout the course, Common Core standards are taught and reinforced as the student learns

how to apply the concepts in real-life situations.

5. The concept of groups, rings, fields and vector spaces are essential building blocks of Modern

algebra and are an integral part of any post graduate course.

SYLLABUS

Unit- I

Learning

Out Comes

Group Theory

Definition of a Group- Some Examples of Groups- Some preliminary Lemmas-

Subgroups-A counting principle- Normal subgroups and Quotient Groups

On successful completion of this unit, students should be able to

1. Learn to concepts of group, Sub group, Normal sub group and Quotient group.

2. Determine whether a given set and binary operation form a group by checking

group axioms.

3. Carry out calculations in quotient groups.

Unit II

Learning

Out Comes

Group Theory Continued..

Homomorphisms – Automorphisms - Cayley”sTheorem- Permutation Groups-

Another counting principle


1. Understand the importance of Homomorphism, automorphisms in group theory

and their applications.

2. Prove properties of homomorphism and understand the connection to normal

subgroups.

3. Define permutation groups and State Cayley’s theorem and its generalization.

Unit III

Learning

Out Comes

Group Theory Continued...

Sylow’s Theorem- Direct products- Finite Abelian Groups.


1. Acquire knowledge of Sylow’s Theorems and finite abelian groups.

2. Apply Sylow theorems to rule out existence of simple groups of certain orders.

3. Express a given finite cyclic group as the direct product of cyclic groups of

prime power order and given two direct products of cyclic groups, determine

whether or not they are isomorphic.

Unit IV

Learning

Ring Theory

Definition and Examples of Rings- Some special classes of Rings-

Homomorphisms- Ideals and Quotient Rings- More Ideals and Quotient Rings-

The Field of Quotients of an Integral Domain.


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Out Comes 1. Learn fundamental concepts of Ring theory that help understand other algebra

courses.

2. Recognize and apply action of rings on sets in geometric and abstract contexts.

3. Understand the applications of integral domain and its properties.

Unit V

Learning

Out Comes

Ring Theory Continued…

Euclidean Rings- A particular Euclidean Ring- Polynomial Rings- Polynomials

over the Rational Field- Polynomial Rings over Commutative Rings.


1. Learn certain Euclidean ring,Polynomial ring and Polynomial ring over

Commutative Ring and their properties.

2. To recognize the reducible & irreducible Polynomial.

3. learn applications of ring theory.

Prescribed

Text Book

Reference

Books

Online

Sources

Topics in Algebra by I. N. Herstein, Second edition, John Wiley & Sons.

1. Modern Algebra by P.B.bhattacharya, S.K.Jain and SR Nagpaul.

2. Group theory byMaroofSamhan and Fadwa Abu MoryafaPublisher Dar Al

Khraiji1st edition 1427 H.

3. A First Course in Abstract Algebra by John B. FraleighPublisher: Pearson, 7th

edition, 2013

4. Contemporary Abstract algebra by J. Gallian, Brooks/Cole Pub Co; 8 edition (13

July 2012).

5. J. A. Gallian, Contemporary Abstract Algebra, Brooks/Cole Cengage Learning, 2010.

1. http://www.supermath.info/AlgebraInotes_2016.pdf

2. http://www-users.math.umn.edu/~garrett/m/algebra/notes/Whole.pdf

3. https://www.cse.iitk.ac.in/users/rmittal/prev_course/s15/notes/complete_notes.pdf.

Course

Out Comes

After completion of this course, students will be able to

1. Understand the concepts of group, subgroups, Normal subgroup and quotient

group and gain knowledge to write some properties of these concepts.

2. Recognize, formulate, classify and solve problems in a mathematical context.

3. Formulate mathematical hypotheses and have an understand of how such

hypotheses can be verified using mathematical methods.

4. Demonstrate knowledge and understanding of fundamental concepts including

groups, subgroups, normal subgroups, homeomorphisms and isomorphism.

5. Students grasp the fundamental principles and theory concerning basic algebraic

structures such as groups, rings, integral domains.

http://www.supermath.info/AlgebraInotes_2016.pdf

http://www-users.math.umn.edu/~garrett/m/algebra/notes/Whole.pdf

https://www.cse.iitk.ac.in/users/rmittal/prev_course/s15/notes/complete_notes.pdf

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Course Code

&Course Name M 102

Real Analysis - I

Objectives1. This course will focus on the proofs of basic theorems of analysis, as appeared in

one variable calculus. Along the way to establish the proofs, many new concepts will be

introduced. Understanding them and their properties are important for the development of the

present and further courses.

2. The course includes axioms of real number systems, convergence of sequences and series,

Continuous functions, uniform continuity and Differentiation and Mean Value theorems.

3. The Fundamental Theorem of Calculus. Series. Power series and Taylor series.

4. Apply mathematical methodologies to open-ended real-world problems

5. Construct rigorous mathematical proofs of basic results in real analysis.

SYLLABUS

Unit I

Learning

Out Comes

Basic Topology: Finite, Countable, and Uncountable Sets, Metric spaces,

Compact sets, Connected sets.

On successful completion of this unit, students will be able to

1. Understand the concept of finite sets, countable and uncountable sets.

2. Describe the real line as a complete, ordered field

3. Use the definitions of convergence as they apply to sequences, series, and

functions.

Unit II

Learning

Out Comes

Numerical Sequences and Series: Convergent sequences, Subsequences, Cauchy

sequences, Upper and Lower limits, Some special sequences, Series, Series of non-

negative terms, number e.


1. Identify the convergence sequences, sub sequences and Cauchy sequences.

2. Understand the concept of upper limit and lower limits, some special sequences

and their applications.

3. Demonstrate an understanding of limits and how they are used in sequences,

series, differentiation and integration

Unit III

Learning

Out Comes

The Root and Ratio tests, Power series, Summation by parts, Absolute

Convergence, Addition and Multiplication of series, Rearrangements.


1. Write the proofs of the root and ratio tests.

2. Apply the power series, summations by parts, absolute convergence.

3. Appreciate how abstract ideas and rigorous methods in mathematical analysis

can be applied to important practical problems.

Unit IV

Learning

Out Comes

Continuity: Limits of Functions, Continuous Functions, Continuity and

Compactness, Continuity and Connectedness, Discontinuities, Monotone

Functions, Infinite Limits and Limits at Infinity.


1. Understand the concepts of limit of functions, continuous functions.

2. Write the simple proofs of the concepts continuity and connectedness, monotone

functions.

3. Identify the infinite limits and limits at infinity.

Unit V

Differentiation:The Derivative of a Real Function, Mean Value Theorems, The

Continuity of Derivatives, L’ Hospital’s Rule, Derivatives of Higher order,

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Learning

Out Comes

Taylor’s theorem, Differentiation of Vector- valued Functions.


1. Determine the derivative real valued functions, mean value theorems.

2. Apply the Mean Value Theorem and the Fundamental Theorem of Calculus to

problems in the context of real analysis

3. Learn applications of the Taylors theorem, differentiation of rector valued

function.

Prescribed

Text

Reference

Books

Online

Source

Principles of Mathematical Analysis by Walter Rudin, International Student

Edition, 3rd Edition, 1985.

1. Mathematical Analysis by Tom M. Apostal, Narosa Publishing House, 2nd

Edition, 1985.

2. Real Analysis by H.L.Royden, Pearson, 4th Ed., 2010.

1. https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf

(Solutions to Principles of Mathematical Analysis by Walter Rudin)

2. https://www.math.lsu.edu/~sengupta/4031f06/IntroRealAnalysNotes.pdf

3. http://www.math.iitb.ac.in/~ars/week1.pdf

4. http://www.rupp.edu.kh/news/documents/A_Course_in_Real_Analysis.pdf

Course Out

Comes

After completing this course students will

1. gain knowledge of concepts of modern analysis, such as convergence,

continuity, completeness, compactness and convexity in the setting of Euclidean

spaces and more general metric spaces

2. develop a higher level of mathematical maturity combined with the ability to

think analytically

3. be able to write simple proofs on their own and study rigorous proofs

4. be able to follow more advanced treatments of real analysis and study its

applications in disciplines such as economics.

5. Demonstrate an understanding of limits and how that are used in sequences,

series and differentiation.

https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf

https://www.math.lsu.edu/~sengupta/4031f06/IntroRealAnalysNotes.pdf

http://www.math.iitb.ac.in/~ars/week1.pdf

http://www.rupp.edu.kh/news/documents/A_Course_in_Real_Analysis.pdf

23

CourseCode

&Course Name M 103

Topology

Objectives1. Topology is the study of properties of spaces invariant under continuous

deformation. For this reason it is often called ``rubber-sheet geometry''

2. This is an introductory course in topology, or the study of shape.

3. Student will to have knowledge on point set topology, topological spaces, Quotient spaces,

Product spaces and metric spaces, sequences, continuity of functions, connectedness and

compactness, homotopy and covering spaces.

4. To explain how to distinguish spaces by means of simple topological invariants (compactness,

connectedness and the fundamental group)

5. Apply your knowledge to solve problems and prove theorems.

SYLLABUS

Unit I

Learning

Out Comes

Topological Spaces: The definition and some examples, interior, closure, and

boundary, Basis and sub basis, Continuity and Topological Equivalence,

Subspaces.

On successful completion of this course, students will be able to

1. Learnthebasic concepts of Topology, basis, subspaces Continuity.

2. Write the proofs of some results on these concepts

3. Apply these concepts in real life problems.

Unit II

Learning

Out Comes

Connectedness: Connected and disconnected spaces, theorems on connectedness,

connected subsets of the real line, applications of connectedness, path connected

spaces, locally connected and locally path connected spaces.


1. Understand the concepts of connectedness and disconnectedness.

2. Write the solutions of some problems on these concepts.

3. Identify the connected spaces and path connected spaces.

Unit III

Learning

Out Comes

Compactness: Compact spaces and subspaces, compactness and continuity,

properties related to compactness, one- pointcompactification, the cantor set.


1. Study the conceptsof compactness and one- point compactification.

2. Write the simple proofs of some results on these concepts.

3. Distinguish the compact and one point compactification.

Unit IV

Learning

Out Comes

Product and Quotient spaces: Finite products, arbitrary products, comparison of

topologies, quotient spaces.


1. Learn the concept of quotient spaces, product spaces.


3. Understand the proofs of these concepts.

Unit V

Learning

Out Comes

Separation properties and Metrization: T0 , T1, and T2 – spaces, regular spaces,

normal spaces, separation by continuous functions, metrization, the stone-cech

compactification.


1. Study the concept of T0 , T1, and T2 – spaces, regular spaces, normal spaces.


3. Learn applications of Urysohn lemma, TheUrysohnmetrization theorem.

24

Prescribed

Text Book

Reference

Text Book

Online

Source

1.Principles of Topology by Fred H. Croom, Cengage learning india private

Limited, Alps building, 1st Floor, %6- Janpath, New Delhi 110001.

1. Topology by James R. Munkers, Second edition, Pearson education Asia – Low

price edition

2. Topology by Dugundji, McGraw-Hill Inc.,US (1 April 1988)

3. Elements of General Topology by Hu, Holden-Day, 1964.

1. https://www.math.colostate.edu/~renzo/teaching/Topology10/Notes.pdf.

2. http://www.pdmi.ras.ru/~olegviro/topoman/part1.pdf.

3. http://www.math.hcmus.edu.vn/~hqvu/n.pdf.

Course Out

Comes


1. know the definitions of standard terms in topology.

2. Students will know how to read and write proofs in topology.

3. Students will know a variety of examples and counterexamples in topology.

4. Students will know about the fundamental group and covering spaces.

5. Students will understand the machinery needed to define homology and co

homology.

6. Students will understand computations in and applications of algebraic

topology.

25

Course Code

&Course Name M 104

Differential Equations

Objectives

1. Describe a collection of methods and techniques used to find solutions to several types of

differential equations, including first order scalar equations, second order linear equations, and

systems of linear equations.

2. Study qualitative techniques for understanding the behavior of solutions.

3. Learn to construct differential equations, corresponding to different ecological or physical

systems.

4. Student will explore: boundary value problems, Sturm-Liouville problems, and Fourier Series.

SYLLABUS

Unit I

Learning

Out Comes

Second order linear differential equations: Introduction-general solution of the

homogeneous equation - Use of a known solution to find another - Homogeneous

equation with constant coefficients - method of undetermined coefficients -

method of variation of parameters


1. Explain the meaning of solution of the homogenous equation.

2. Solve the homogenous second order equation with constant coefficient.

3. Applies the method of undetermined coefficient and method of variation of

parameters to find the solution of second order linear differential equation with

variable coefficients.

Unit II

Learning

Out Comes

Oscillation theory and boundary value problems: Qualitative properties of

solutions - The Sturm comparison theorem - Eigen values, Eigen functions and the

vibrating string.


1. Solve the qualitative properties of solutions.

2. Expresses the Sturm comparison theorem.

3. Solve the Eigen values, eigen functions and the vibrating string.

Unit III

Learning

Out Comes

Power series solutions: A review of power series-series solutions of first order

equations-second order linear equations - ordinary points-regular singular points


1. Solve the power series solutions of first order equations.

2. Solve the power series solutions of second order equations with ordinary points.

3. Solve the regular singular point.

Unit IV

Learning

Out Comes

Systems of first order equations: Linear systems - Homogeneous linear systems

with constant coefficients


1. Learn the system of first order equations.

2. Determines the types of linear differential equation systems.

3. Solve the homogenous linear systems with constant coefficient.

Unit V

Learning

Out Comes

Existence and Uniqueness of solutions - successive approximations - Picard’s

theorem - Some examples


1. Learn the existence and uniqueness of solutions.

2. Apply the method of successive approximations.

26

3. Expresses the Picard’s theorem of differential equations.

Prescribed

Text Book

Reference

Books

Online

Source

1. George F. Simmons, Differential Equations, Tata McGraw-Hill Publishing

Company Limited, New Delhi.

1. Ordinary Differential Equations and stability theory by S. G. Deo and V.

Raghavendra, TATA Mc Graw Hill Ltd, 1980.

2. Theory of Ordinary Differential Equations by Coddington and Levinson,

Krieger Pub Co (June 1984).

1. https://www.math.ust.hk/~machas/differential-equations.pdf.

2. http://www.math.toronto.edu/selick/B44.pdf.

3. https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf.

Course Out

Comes


1.The study of Differential focuses on the existence and uniqueness of solutions

and also emphansizes the rigorous justification of methods for approximating

solutions in pure and applied mathematics.

2.It plays an important role in modelling virtually every physically technical or

biological process from celestial motion to bridge design to interactions between

neurons.

3.Theory of differential equations is widely used in formulating many fundamental

laws of physics and chemistry.

4. Theory of differential equation is used in economics and biology to model the

behavior of complex systems.

5. Differential equations have a remarkable ability to predicts the world around us.

They can describe exponential growth and decay population growth of species or

change in investment return over time.

27

Course Code

&Course Name M105

Linear Algebra

Objectives

1. To make the students familiar with principles and techniques of Linear algebra and their

applications.

2. To effective communicative ideas and explain procedures.

3. To effectively interpret results and solutions in written.

4. To identify and develop system equation models.

5. To make effective techniques in Linear algebra.

SYLLABUS

Unit I

Learning

Out Comes

The Elementary Canonical Forms: Introduction-Characteristic Values-Annihilating

Polynomials-Invariant Subspaces


1. Determine relationship between coefficient matrix invertibility and solutions to

a system of linear equations and the inverse matrices.

2. Determine the concept of Annihilating Poly, Invariant Subspaces.

3. Learn Applications of Annihilating Poly, Invariant Subspaces.

Unit II

Learning

Out Comes

Simultaneous Triangulation-Simultaneous Diagonalization-Direct-sum

Decompositions,Invariant Direct Sums- Derive The Primary Decomposition

Theorem.


1. Understand linear independence and dependence.

2. Find basis and dimension of a vector space, and understand change of basis.

3. Determine the Applications of Primary Decompositon.

Unit III

Learning

Out Comes

Cyclic subspaces and annihilators, Derive Cyclicdecompositions and the rational

form, Evaluate the Jordon form, Computation of Invariant factors.


1. Understand the concept of Cyclic subspaces, annihilators.

2. Derive Cyclicdecompositions and the rational form.

3. Learn the systems of linear equations using various methods.

Unit IV

Learning

Out Comes

Semi-Simple operators and Bilinear forms.


1.Understand the concept of Semi-Simple operators,

2. Understand the concept of Bilinear forms.

3. Determine the applications of different operators.

Unit V

Learning

Out Comes

Evaluate Symmetric Bilinear form, Skew Symmetric Bilinear form, and Group

Preserving Bilinear form.


1. Understand the concept of Symmetric and Skew Bilinear form.

2. Understand the concept of Group Preserving Bilinear form.

3. Determine the applications of Bilinear forms.

Prescribed

Text Book

1. Linear Algebra second edition By Kenneth Hoffman and Ray Kunze, Prentice-

Hall of India Private Limited, New Delhi-110001, 2002

28

Reference

Books

Online

Source

1. A. RamachandraRao and P. Bhimsankaram. Linear Algebra, Hindustan Book

Agency; 2nd Revised edition edition (15 May 2000.

2. S. Kumaresan-Linear Algebra, Prentice Hall India Learning Private Limited;

New title edition (2000).

1. http://joshua.smcvt.edu/linearalgebra/book.pdf

2.http://linear.ups.edu/download/fcla-3.40-tablet.pdf

3.http://omega.albany.edu:8008/mat220/LAbook.pdf

4.http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_alg_Lang.pdf

Course

Out Comes


1. Use computational techniques and algebraic skills essential for the study of

systems of linear equations, matrix algebra, vector spaces, eigenvalues and

eigenvectors, orthogonality and diagonalization.

2. Evaluate determinants and use them to discriminate between invertible and non-

invertible matrices.

3. Identify linear transformations of finite dimensional vector spaces and compose

their matrices in specific bases.

4. Use visualization, spatial reasoning, as well as geometric properties and

strategies to model, solve problems, and view solutions, especially in R2 and R3,

as well as conceptually extend these results to higher dimensions.

5. Critically analyze and construct mathematical arguments that relate to the study

of introductory linear algebra.

http://joshua.smcvt.edu/linearalgebra/book.pdf

http://linear.ups.edu/download/fcla-3.40-tablet.pdf

http://omega.albany.edu:8008/mat220/LAbook.pdf

http://www.math.nagoya-u.ac.jp/~richard/teaching/f2014/Lin_alg_Lang.pdf

29

M.Sc Mathematics FIRST YEAR – SECOND SEMESTER

CourseCode&Cours

e Name M 201

Algebra - II

Objectives1. As a second course in algebra the objective of this course is to have a complete

understanding of fields and linear transformations.

2. The concept of Galois theory in fields is central to theory of equations and is a must for all

mathematics students.

3. Student will explore: Fields, linear transformations, finite fields.

4. Appraise how to use the computer skills and library.

5. The knowledge on this course will provide the basis for further studies in advanced algebra like

commutative algebra, linear groups, modules etc., which forms the basics of higher mathematics.

SYLLABUS

Unit I

Learning

Out Comes

Fields: Extension Fields- The Transcendence of e – Roots of Polynomials

(Chapter 5 sections 5.1 -5.4)

On successful completion of this Unit, students should be able to :

1. Learn to concepts of Fields, extension fieldsandTranscendence numbers.

2. Determine whether a given set and binary operation form a filed by checking

field axioms.

3. Carry out calculations in polynomials roots.

Unit II

Learning

Out Comes

Fields: Construction with Straightedge and Compass, more about roots.

(Chapter 5 sections 5.1 -5.4)

After studying this unit, students should be able to :

1. Understand importance of More about roots of polynomials,

2. Explain what is meant by a symmetry of a plane figure.

3. Describe the symmetries of some bounded three-dimensional figures.

4. Use the straightedge and compass to identify and construct examples of the

reducible and irr-reducible polynomials.

Unit III

Learning

Out Comes

Fields:The elements of Galois Theory- Solvability by Radicals- Galois Groups

over the Rationals.

(Chapter 5 sections 5.5- 5.8)

After studying this unit, the student is expected to be able to:

1. Acquire knowledge of Galois Theory

2. Apply Galois Theoryto rule out existence of simple extension of fixed fields.

3. Determine the elementary symmetric functions of symmetric polynomials.

Unit IV

Learning

Out Comes

Finite Fields: Wedderburn’s Theorem on Finite Division Rings.(Chapter 7

sections 7.1 , 7.2)

Upon successful completion of this unit, students will be able to:

1.Learn concept of Wedderburn’s Theorem

2. Give examples of skew field and prove their simplicity.

3. Prove properties of finite division rings and understand the connection to fields

Unit V

Finite Fields: A Theorem of Frobenius- Integral Quaternions and the Four-Square

Theorem.(Chapter 7 sections 7.3, 7.4)

30

Learning

Out Comes

Upon successful completion of this unit, students will be able to:

1. Prove the Frobenius theorem

2. Determine the Lagrange identity and left division algorithm.

3. Prove the Four-Square Theorem and their applications.

Prescribed

Text Book

Reference

Books

Online

Source

Topics in Algebra: I. N. Herstein , Second edition, John Wiley & Sons.

1. Modern Algebra by P.B.bhattacharya, S.K.Jain and SR Nagpaul.

2. Group theory byMaroofSamhan and Fadwa Abu MoryafaPublisher Dar Al

Khraiji1st edition 1427 H.

3. A First Course in Abstract Algebra by John B. FraleighPublisher: Pearson, 7th

edition, 2013

4. Contemporary Abstract algebra by J. Gallian, Brooks/Cole Pub Co; 8 edition (13

July 2012).

1. http://www.supermath.info/AlgebraInotes_2016.pdf

2. http://www-users.math.umn.edu/~garrett/m/algebra/notes/Whole.pdf

3. https://www.cse.iitk.ac.in/users/rmittal/prev_course/s15/notes/complete_notes.pdf

Course

Out Comes


1. Recognize technical terms and appreciate some of the uses of algebra

2. Demonstrate insight into abstract algebra with focus on axiomatic theories

3. Apply algebraic ways of thinking

4. Demonstrate knowledge and understanding of fundamental concepts including

fields, extension fields and finite fields.

5. Students grasp the fundamental principles and theory concerning basic algebraic

structures such as fields and extension fields

http://www.supermath.info/AlgebraInotes_2016.pdf

http://www-users.math.umn.edu/~garrett/m/algebra/notes/Whole.pdf

https://www.cse.iitk.ac.in/users/rmittal/prev_course/s15/notes/complete_notes.pdf

31

Course Code

&Course Name M 202

Real Analysis - II

Objectives1. This course will focus on the proofs of basic theorems of analysis, as appeared in

one variable calculus. Along the way to establish the proofs, many new concepts will be

introduced. Understanding them and their properties are important for the development of the

present and further courses.

2. The course includes axioms of real number systems, uniform convergence of sequences and

series, Continuous functions, uniform continuity and Differentiation and Mean Value theorems.

3. The Fundamental Theorem of Calculus. Series. Power series and Taylor series.

4. Investigate the consequences of different types of convergence for sequences and series of

functions, the interchange of limits with other operations such as integrals and derivatives

5. Also develop some of the calculus of functions of several variables including the general

inverse and implicit function theorems.

SYLLABUS

Unit I

Learning

Out Comes

Riemann-Stieltjes Integral: Definition and existence of the Riemann Stieltjes

Integral, Properties of the Integral, Integration and Differentiation, the fundamental

theorem of calculus – Integral of Vector- valued Functions.

(Chapter 6 of the prescribed text book)


1.Determine the Riemann integrability of a bounded function and prove a selection

of theorems concerning integration.

2. Understand the applications of Riemann-Stieltjes Integral

3. Apply these concepts in real world problems.

Unit II

Learning

Out Comes

Sequences and Series of the Functions: Discussion on the Main Problem, Uniform

Convergence, Uniform Convergence and Continuity, Uniform Convergence and

Integration, Uniform Convergence and Differentiation

(First five sections of Chapter 7 of the prescribed text book)


1. Understand the concepts of Uniform Convergence, Uniform Convergence and

Continuity, Uniform Convergence and Integration

2. Recognize the difference between point wise and uniform convergence of a

sequence of functions

3. Illustrate the effect of uniform convergence on the limit function with respect to

continuity, differentiability, and integrability.

Unit III

Learning

Out Comes

Equicontinuous families of Functions, the Stone-Weierstrass Theorem.

(6th& 7thsections of Chapter 7 of the text book)

Power Series: (A section in Chapter 8 of the text book)


1. Understand the concepts of Equicontinuous families of Functions, the Stone-

Weierstrass Theorem.

2. Identify the series and power series.

3. Studying applications of Equicontinuous families of Functions, the Stone-

Weierstrass Theorem.

Unit IV

Functions of Several Variables: Linear Transformations, Differentiation, The

Contraction Principle, The Inverse Function theorem. (First Four sections of

chapter 9 of the text book)

32

Learning

Out Comes


1. Illustrate the convergence properties of power series.

2. Studying the concept of contraction principle, properties and their applications.

3. To apply the acquired knowledge in signals and Systems, Digital Signal

Processing. Etc

Unit V

Learning

Out Comes

Functions of several variables Continued: The Implicit Function theorem, The

Rank theorem, Determinates, Derivatives of Higher Order, Differentiation of

Integrals.

(5th to 9th sections of Chapter 9 of the text book)


1.Studying the concept of implicit function theorem and its applications.

2. Understand the concepts of Rank theorem, Determinates, Derivatives of Higher

Order, Differentiation of Integrals.

3. Learn to apply these concepts to real problems

Prescribed

Text Book

Reference

Books

Online

Source

Principles of Mathematical Analysis by Walter Rudin, International Student

Edition, 3rd Edition, 1985.

1. Mathematical Analysis by Tom M. Apostal, Narosa Publishing House, 2nd

Edition, 1985.

2. Real Analysis by H.L.Royden, Pearson, 4th Ed., 2010.

1. https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf

(Solutions to Principles of Mathematical Analysis by Walter Rudin)

2. https://www.math.lsu.edu/~sengupta/4031f06/IntroRealAnalysNotes.pdf

3. http://www.math.iitb.ac.in/~ars/week1.pdf 4. http://www.rupp.edu.kh/news/documents/A_Course_in_Real_Analysis.pdf

Course

Out Comes


1. Have sound mathematical knowledge. They have a profound overview of the

contents of fundamental mathematical disciplines and are able to identify their

correlations.

2. Write simple proofs of the results and study the applications.

3. Demonstrate an understanding of limits ad how that is used in sequences of

functions, series of functions and differentiation.

4. Construct rigorous mathematical proofs of basic results in real analysis.

5. Appreciate how abstract ideas and regions methods in mathematical analysis can

be applied to important practical problems.

https://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf

https://www.math.lsu.edu/~sengupta/4031f06/IntroRealAnalysNotes.pdf

http://www.math.iitb.ac.in/~ars/week1.pdf

http://www.rupp.edu.kh/news/documents/A_Course_in_Real_Analysis.pdf

33

Course Code

&Course Name M 203

Probability

Objectives

1.This course provides an elementary introduction to probability and statistics with applications.

2. Student will explore: basic probability models; combinatorics; random variables; discrete and

continuous probability distributions; statistical estimation and testing; confidence intervals; and

an introduction to linear regression.

3. Be able to use the addition law and be able to compute the probabilities of events using

conditional probability and the multiplication law.

4. Be able to use new information to revise initial (prior) probability estimates using Bayes'

theorem.

5. Apply problem-solving techniques to solving real-world events.

SYLLABUS

Unit I

Learning

Out Comes

Combinatorial analysis: Introduction,The Basic Principle of counting,

Permutations, Combinations, Multinomial Coefficients,, The number of integer

solutions of equations. Axioms of Probability: Introduction,Sample space and

events, Axioms of Probability, Some Simple Propositions, Sample Spaces Having

Equally Likely Outcomes, Probability as a Continuous Set Function.

Section 1.1 to 1.6 of Chapter 1 and 2.1 to 2.6 of Chapter 2


1.Understanding the basic concepts of basic principle of counting, Permutations,

Combinations, Multinomial Coefficients, Sample space and events, Axioms of

Probability, Some Simple Propositions

2. Understand the meaning of probability and probabilistic experiment

3. Obtain an understanding of the role probability information plays in the decision

making process.

Unit II

Learning

Out Comes

Conditional Probability and Independence:Introduction, Conditional Probabilities,

Bayes’s Formula, Independent Events, P(·|F) Is a Probability.

Section 3.1 to 3.5 of chapter 3


1. Understand the meaning of conditional probability, conditioning and reduced

simple space.

2. Study applications of Conditional Probabilities, Bayes’s Formula.

3. Distinguish independent events and dependent events.

Unit III

Learning

Out Comes

Random Variables: Random Variables, Discrete Random Variables,Expected

Value, Expectation of a Function of a Random, The Bernoulli and Binomial

Random,Properties of Binomial Random Variables,Computing the Binomial

Distribution Function.

Section 4.1 to 4.6 of Chapter 4


1. Demonstrate understanding the random variable, expectation, variance and

distributions.

2. Partially characterize a distribution using a expected value, variance and

moments.

3. Derive distributive functions of a random variable.

Unit IV

Poisson Random, Computing the Poisson Distribution Function, Expected Value

of Sums of Random Variables, Properties of the Cumulative Distribution Function.

34

Learning

Out Comes

Section 4.7 & 4.9 to 4.10 of Chapter 4


1. Understand the concepts of Poisson Random, Computing the Poisson Distribution

Function, Expected Value of Sums of Random Variables, Properties of the

Cumulative Distribution Function.

2. Calculate various moments of common random variables including at least

means, variances and standard deviations.

3. Identify important types of distribution functions such as Poisson distribution

function, Binomial distribution function and Cumulative Distribution Function.

Unit V

Learning

Out Comes

Limit Theorems:Introduction, Chebyshev’s Inequality and the Weak Law of Large

Numbers, TheCentralLimitTheorem,The Strong Law of Large Numbers.Additional

Topics in Probability, The Poisson Process, MarkovChains.

Section 8.1 to 8.4 Chapter 8 and 9.1 to 9.2 Chapter 9


1. Understand limit theorems such as Weak Law of Large Numbers,

TheCentralLimitTheorem,The Strong Law of Large Numbers

2. Learn applications of law of large numbers and the central limit theorem and

how these concepts are used to model various random phenomena.

3. Extend the concept of a random variable to a random process and understand the

basic concept of random process.

Prescribed

Text Book

Reference

Books

Online

Source

A First course in Probability by Sheldon Ross, Eight Edition 2010.

Pearson Publications

1. Modern probability theory by B.R. Bhat, Wiley, 1985.

2. Probability and measure by P. Billingsley, Wiley, 1986.

3. A graduate course in probability theory by H.G.Tucker, 1967, (AP)

1. http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf

2. https://www.stat.berkeley.edu/~aldous/134/gravner.pdf

3. http://www.iiserpune.ac.in/~ayan/MTH201/Sahoo_textbook.pdf.

Course

Out Comes

Upon completion of this course, the student will be able to

1. Compute the probabilities of composite events using the basic rules of probability.

2. Demonstrate understanding the random variable, expectation, variance and

distributions.

3. Compute the sample mean and sample standard deviation of a series of independent

observations of a random variable.

4. Apply the concepts of multiple random variables to engineering applications.

5. Analyze the correlated data and fit the linear regression models.

http://julio.staff.ipb.ac.id/files/2015/02/Ross_8th_ed_English.pdf

https://www.stat.berkeley.edu/~aldous/134/gravner.pdf

35

Course Code

&Course Name M 204

COMPLEX ANALYSIS

Objectives1. This course is aimed to provide an introduction to the theories for functions of a

complex variable.

2. Student will explore: Elementary properties and examples of analytic functions: Power series-

Analytic functions- Analytic functions as mappings.

3. It begins with quick review on the exploration of the algebraic, geometric and topological

structures of the complex number field.

4. The concepts of analyticity and mapping properties of function of a complex variable will be

illustrated.

5. Complex integration and complex power series are presented.

SYLLABUS

Unit I

Learning

Out Comes

Elementary properties and examples of analytic functions: Power series- Analytic

functions- Analytic functions as mappings, Mobius transformations

($1, $2, $3 of chapter-III of prescribed text book)


1. Determine the elementary properties and examples of analytic functions.

2. Evaluate the power series analytic functions as mappings.

3. Understanding and evaluate analytic functions and Mobius transformation

functions.

Unit II

Learning

Out Comes

Complex Integration: Riemann- Stieltjes integrals- Power series representation of

analytic functions

($1, $2 of chapter-IV of prescribed text book)


1.Understanding the concept of Riemann-Stieltjescomplex integrals and power

series with applications.

2. Expresses the cauchy’s integral formula.

3. Determine the Riemann stieltjes integrals with applications.

Unit III

Learning

Out Comes

Zeros of analytic functions- The index of a closed curve, Cauchy’s theorem and

integral formula.

($3 $4, $5 of chapter-IV of prescribed text book).


1. Explaining the concepts, proof of Cauchy’s theorem.

2. Determine the zeros of analytic functions.

3. Expresses the Cauchy’s theorem, Cauchy’s integral formula and Moreara’s

theorem.

Unit IV

Learning

Out Comes

The homotophic version of Cauchy’s theorem and simple connectivity- Counting

zeros; the open mapping theorem.

($6, $7 of chapter-IV of prescribed text book)


1. Understand the concept of homotophic version of Cauchy’s theorem and simple

connectivity- Counting zeros; and proof of the open mapping theorem.

2. Determine the concept of homotopic version of Cauchy’s theorem and simple

connectivity.

3. Evaluate the counting zeros of analytic functions.

Unit V Singularities: Classifications of singularities- Residues- The argument principle.

36

Learning

Out Comes

($1, $2, $3 of chapter-V of prescribed text book)


1.Classification of singularities.

2. Evaluate the Residue and expresses the residue theorem.

3. Learn the concept singularities and their applications.

Prescribed

Text Book

Reference

Books

Online

Resource

Functions of one complex variables by J.B.Conway : Second edition, Springer

International student Edition, Narosa Publishing House, New Delhi.

1. Complex Variable and Applications by R.V. Churchill and J. W. Brown, Tata

McGraw Hill, 2008.

2. Complex Analysis by L.V.Ahlfors, Tata McGraw Hill, 1979.

3. Foundation of Complex Analysis by S. Ponnuswamy,Narosa Publishing House,

2007.

4. Complex Variables: Theory and Applications by H.S. Kasana, PHI, 2006. 1. http://www.maths.lth.se/matematiklu/personal/olofsson/CompHT06.pdf.

2.https://www.if.ufrj.br/~tgrappoport/aulas/metfis1/complex.pdf

3. http://math.sfsu.edu/beck/papers/complex.pdf.

Course

Out Comes


1. Analyze limits and continuity for functions of complex variables, understand

about the Cauchy-Riemann equations, analytic functions, entire functions

including the fundamental theorem of algebra,

2. Evaluate complex contour integrals and apply the Cauchy integral theorem in its

various versions, and the Cauchy integral formula,

3. Analyze sequences and series of analytic functions and types of convergence,

4. Represent functions as Taylor and Laurent series, classify singularities and

poles, find residues and evaluate complex integrals using the residue theorem’

5. Understand the conformal mapping and apply real world problems

37

Course Code &

Course Name M 205

Graph Theory

Objectives

1. Graphs are used to model networking problems in physical and biological sciences etc.

2. As an essential tools in computer and information sciences, the concepts in Graph Theory

address problems of social media, linguistics, chemical bonds, computational neuro science,

market and financial analysis, communication system, data organisation, flows and links.

3. Graphs are mathematical constructions used to describe collections of objects some pairs of

which are related to each other. For example a family tree is a collection of people in which some

are related to others by parentage.

4. To understand and apply the fundamental concepts in graph theory.

5. To apply graph theory based tools in solving practical problems.

SYLLABUS

Unit I

Learning

Out Comes

Basic concepts, Isomorphism, Euclidian and Hamilton Graphs


1. Learn the concepts of connectedness, walk, path circuits.

2.Write examples of connected graphs, walk, path circuits Euler graph, etc.

3. Apply these concepts in real life problems

Unit II

Learning

Out Comes

Trees, Properties of Trees, Spanning Trees, Connectivity and Separability,

Network flows.


1. Determine the concept of Graph theory and different types of graph.

2. Understand the concept of Fundamental circuits and cut-sets

3. Describe the network flows and isomorphism

Unit III

Learning

Out Comes

Planar graphs, Kuratowski’s two graphs, Different representations of planar

graphs, Detection of Planarity, Geometric and Combinational Duals of a graph,

Vector spaces of a Graph.


1. Will be able to learn the concept of fundamental cycles spanning concept

2. Describe the concept of geometric dual, spanning concept.

3. Solving the application of fundamental cycles spanning concept.

Unit IV

Learning

Out Comes

Matrix representation of graphs, Incidence and circuit matrices of a graph,

Fundamental Circuit matrix, Application to a Switching network, Cut set and Path

Matrices, Adjacency matrices, Directed Graphs, Trees with directed Edges,

Incidence and adjacency matrix of a digraph.


1. Determine the concept of Coloring problems.

2. Describe the concept of chromatic number.

3. Solve the application of four color problem.

Unit V

Learning

Coloring, Covering and Partitioning, Chromatic number, Chromatic Partitioning,

Chromatic polynomial, Matchings, Coverings, The form color problem,

Applications of graph theory inOperations Research.


38

Out Comes 1. Describe the concept of Directed graphs.

2. Determine the concept of binary relations.

3. Solve the concept of Euler digraphs.

Prescribed

Text Book

Reference

Books

Online

Source

Graph Theory and its Application to Engineering and Computer Science by

NarsingDeo, PHI, 1979.

1. Graph Theory by F. Harary, Addison Wesley Publishing company, 1969.

2. Introduction to Graph Theory by R. J. Wilson, Longman Group Ltd., 1985 .

3.Graph Theory with applications by Bond JA and Murthy USR, North Holland,

New York.

4.Discrete Mathematics and Graph Theory (by BhavanariSatyanarayana and

KunchamSyam Prasad), Second Edition (2014), PHI Learning Private Limited,

New Delhi, ISBN : 978-81-203-4948-3.

5.Mathematical Foundation of Computer Science (by Dr.BhavanariSatyanarayana,

Dr.TumurukotaVenkata Pradeep Kumar, Dr. Shaik Mohiddin Shaw), BS

Publications, Hyderabad, 2016 (ISBN: 978-93-83635-81-8) and CRC Press,

England, 2019 (ISBN-13: 978-0-367-3681-0).

1. http://www.zib.de/groetschel/teaching/WS1314/BondyMurtyGTWA.pdf.

2. http://math.tut.fi/~ruohonen/GT_English.pdf.

3. https://www.maths.ed.ac.uk/~v1ranick/papers/wilsongraph.pdf.

Learning

Out Comes


1. Key concepts from the text and questions arising

2. Investigation of questions posed for seminar discussion;

3. Occasional presentation by students of key items from the SYLLABUS;

4. Strategies for thinking about graph theory and about mathematics generally;

5. Proofs in graph theory.

6. Mathematical writing.

Course

Out Comes

After completion of this course, students will be able

1. Demonstrate knowledge of the SYLLABUS material;

2. Write precise and accurate mathematical definitions of objects in graph theory;

3.Use mathematical definitions to identify and construct examples and to

distinguish examples from non-examples;

4. Validate and critically assess a mathematical proof;

5.Use a combination of theoretical knowledge and independent mathematical

thinking in creative investigation of questions in graph theory;

6. Reason from definitions to construct mathematical proofs;

7. Write about graph theory in a coherent and technically accurate manner.

39

M.Sc Mathematics SECOND YEAR – THIRD SEMESTER

Course Code

&Course Name M 301

FUNCTIONAL ANALYSIS

Objectives1. The objective of the course is the study of the main properties of bounded operators

between Banach and Hilbert spaces.

2. The basic results associated to different types of convergences in normed spaces and the

spectral theorem and some of its applications.

3. The course gives an introduction to functional analysis, which is a branch of analysis in which

one develops analysis in infinite dimensional vector spaces.

4. The central concepts which are studied are normed spaces with emphasis on Banach and

Hilbert spaces, and continuous linear maps (often called operators) between such spaces.

5. Spectral theory for compact operators is studied in detail, and applications are given to integral

and differential equations.

SYLLABUS

UNIT I

Learning

Out Comes

Banach spaces: The definition and some examples, continuous linear

transformation, The Hahn-Banach theorem, the natural imbedding of N in N**.

(Sections 46 to 49 of Chapter 9 of the text book 1)

On successful completion of this unit,students will be able to

1. understand the concept of Banach space and applications of some results on

Banach spaces. 2. Analyze the normed linear spaces, Banach space and Dual spaces

3. Role of completeness through the Baire category theorem and its consequences

for operators on Banach spaces.

UNIT II

Learning

Out Comes

The open mapping theorem, The conjugate of an operator, Hilbert spaces: The

definition and some simple properties, orthogonal complements.

(Sections 50, 51 of Chapter 9 & Sections 52, 53 of Chapter 10 of the text book 1.)


1. Understand the open mapping theorem and its applications. 2. Understand inner product spaces, orthogonality and Hillbert spaces.

3. State and apply the Banach Isomorphism Theorem and Closed Graph Theorem

to determine whether operators are bounded.

UNIT III

Learning

Out Comes

Orthonormal sets, The conjugate space H*, the adjoint of an operator, Self-

adjointoperators.

(Sections 54 to 57 of Chapter 10 of the text book 1.)


1.Understand the concept of Orthonormal sets, adjoint operators and its

applications

2. Have a demonstrable knowledge of the properties of a Hilbert space, including

orthogonal complements, orthonormal sets, complete orthonormal sets together

with related identities and inequalities

3.Familiar with the theory of linear operators on a Hilbert space, including adjoint

operators, self-adjoint and unitary operators with their spectra.

UNIT IV

Normal and Unitary operators, Projections, Finite-dimensional spectral theory:

Matrices.

(Sections 58, 59 of Chapter 10 & Section 60 of Chapter 11 of the text book 1.)

40

Learning

Out Comes


1.Understand the concept of normal and unitary operators and its

applications 2. Determine the finite and infinite dimensional spaces.

3. Determine whether linear operators are continuous, invertible, self-adjoint,

compact etc, and determine adjoints.

UNIT V

Learning

Out Comes

Determinants and the spectrum of an operator, the spectral theorem.A survey

of the situation.

(Sections 61 to 63 of Chapter 11 of the text book 1.)


1. Analyze the concept of spectrum of an operator and the real world

applications on this. 2. Apply linear operators in the formulation of differential and integral equations.

3. Define the spectrum of an operator, and derive basic properties

Prescribed

Text Book

Reference

Books

Online

Source

Introduction to Topology and Modern Analysis by G. F. Simmons, McGraw Hill

Book Company. Inc-International student edition

1. Functional Analysis A First Course by M.Thamban Nair

2. A Course in Functional Analysis by Conway, John B. Springer.

2. Functional Analysis by B. V. Limaye, Willey Eastern Limited, Bombay 1981.

3. First Course in Functional Analysis, C. Goffman and George Pedrick,

Prentice Hall of India Private Limited, New Delhi-110001.

4. E. Kreyszig, Introductory Functional Analysis with applications, Wiley Eastern,

1989.

5. Functional Analysis by Bachmen and Narici, Dover Publications Inc.; 2nd

edition edition (28 March 2003).

1. http://www.math.kit.edu/iana1/lehre/funcana2012w/media/fa-lecturenotes.pdf

2. https://users.math.msu.edu/users/jeffrey/920/920notes.pdf

3. https://www.mimuw.edu.pl/~aswiercz/AnalizaF/lecture.pdf

Course

Outcomes

1. The fundamental properties of normed spaces and of the transformations

between them.

2. To be acquainted with the statement of the Hahn-Banach theorem and its

corollaries.

3. To understand the notions of dot product and Hilbert space.

4. To apply the spectral theorem to the resolution of integral equations

Sturm-Liouville problems.

5. Demonstrate capacity for mathematical reasoning through analysing proving

and explaining concepts from functional analysis.

http://www.math.kit.edu/iana1/lehre/funcana2012w/media/fa-lecturenotes.pdf

https://users.math.msu.edu/users/jeffrey/920/920notes.pdf

https://www.mimuw.edu.pl/~aswiercz/AnalizaF/lecture.pdf

41

Course Code

&Course Name M 302

NUMBER THEORY

Objectives

1. The objective of the course is the study of the main properties of Primes, Divisibility,

Fundamental Theorem of Arithmetic, Greatest Common Divisor and Euclidean Algorithm

2. The basic results associated to Euler’s summation formula, asymptotic formulas and the

average order of d(n), ( )n , (n) and its applications.

3. The course gives an introduction to strong background of the Mobius function and the partial

sums of the Mobius function with their applications.

4. The central concepts which are studied properties of congruences, resudueclasses, complete

residue systems, linear congruences and reduced residue systems

5. To develop their skills in the programming of Characters in finite abelian groups, the

orthogonality relations,Dirichlet characters.

SYLLABUS

Unit I

Learning

Out Comes

Arithmetical Functions And Dirichlet Multiplication: Introduction- The Mobius

function function (n) – The Euler totient function (n)- A relation connecting

and - A product formula for (n)- The Dirichlet product of arithmetical

functions- Dirichlet inverses and the Mobius inversion formula- The Mangoldt

function (n)- multiplicative functions- multiplicative functions and Dirichlet

multiplication- The inverse of a completely multiplicative function-Liouville’s

function ( )n - The divisor functions ( )n . Chapter-2:- Articles 2.1 to 2.14

On successful completion of this Unit, students should be able to :

1. Prove properties of Mobius functions, Euler totient function and understand the

connection to

and

.

2. Determine the dirichlet product of arithmetical functions

3. Explian what is meant by a multiplicative and completely multiplicative of a

arithmetical functions.

Unit II

Learning

Out Comes

Averages of arithmetical functions: Introduction- The big oh notation. Asymptotic

equality of functions- Euler’s summation formula- Some elementary asymptotic

formulas-The average order of d(n)- The average order of the divisor functions

( )n - The average order of (n). The partial sums of a Dirichlet product-

Applications to (n) and (n)- Another identity for the partial sums of a

Dirichletproduct.Chapter -3:- Articles 3.1 to 3.7

After studying this unit, students should be able to:

1.Apply the Euler summation formula in asymptotic equalities and find the

average order of arithmetical functions

2. Determine the partial sums of dirichlet product.

3. Express applications of (n) and (n).

Unit III

some elementary theorems on the distribution of prime numbers:Introduction-

Chebyshev’s functions ( )x and ( )x - Relations connecting ( )x and ( )x -

Some equivalent forms of the prime number theorem-Inequalities for ( ) and pnn -

Shapiro’s Tauberian theorem- Applications of Shapiro’s theorem- An asymptotic

formula for the partial sums (1/ )p x

p

- The partial sums of the Mobius function –

The partial sums of the Mobius function. Chapter -3:- Articles 3.10 &3.11 and

Chapter-4:- Articles 4.1 to 4.9

42

Learning

Out Comes


1. Acquire knowledge of prime number theorem and Shapiro’s Tauberian theorem

with their Applications.

2. Understand the connection between chebyshev’s functions

3. Apply the Shapiro’s Tauberian theorem in asymptotic formulas.

Unit IV

Learning

Out Comes

CONGRUENCES: Definition and basic properties of congruences- Resudue

classes and complete residue systems- Linear congruences- Reduced residue

systems and the Euler- Fermat theorem- Polynomial congruences modulo p.

Lagrange’s theorem- Applications of Lagrage’s theorem- Simultaneous linear

congruences. The Chinese remainder theorem- Applications of the Chinese

remainder theorem- Polynomial congruences with prime power moduli. Chapter -

5:- Articles 5.1 to 5.9

On successful completion of this unit, students should be able to:

1.Determine thecongruences,

2. Prove theEuler- Fermat theorem, Lagrage’stheoremand Chinese remainder

theorem.

3. Give examples of linear congruences and their RRS properties.

Unit V

Learning

Out Comes

FINITE ABELIAN GROUPS AND THEIR CHARACTERS:

Characters of finite abelian groups- The character group- The orthogonality

relations- for characters- Dirichlet characters- Sums involving Dirichlet characters-

The nonvanishing of

L(1, ) for real nonprincipal .

DIRICHLET’S THEOREM ON PRIMES IN ARITHMETIC

PROGRESSIONS:

Introduction- Dirichlet’s theorem for primes of the form 4n-1 and 4n+1- The plan

of the proof of Dirichlet’s theorem- Proof of Lemma 7.4- Proof of Lemma 7.5-

Proof of Lemma 7.6- Proof of Lemma 7.7- Proof of Lemma 7.8- Distribution of

primes in arithmetic progressions.

Chapter 6:- Articles 6.5 to 6.10 and Chapter 7 :- 7.1 to 7.9

After studying this unit, students will be able to

1. Find the Characters of finite abelian groups

2. Prove dirichlet’s theorem and their properties.

3.Express orthogonality relations of characters and dirichlet characters.

Prescribed

Text Book

Reference

Books

Online

Source

Introduction to Analytic Number Theory- By T.M.APOSTOL-Springer Verlag-

New York, Heidalberg-Berlin-1976.

1. A Course in Number Theory and Cryptography Neal Koblitz, Graduate Texts in

Mathematics, New-York: Springer-Verlag, 1987.

1. http://homepages.warwick.ac.uk/staff/J.E.Cremona/courses/MA257/ma257.pdf

2. http://www2.math.uu.se/~astrombe/talteori2016/lindahl2002.pdf

Course

OutComes

After studying this course, you should be able to:

1. find quotients and remainders from integer division

2. apply Euclid’s algorithm and backwards substitution

3. understand the definitions of congruences, residue classes and least residues

4. add and subtract integers, modulo n, multiply integers and calculate powers,

modulo n

5. determine multiplicative inverses, modulo n and use to solve linear congruences.

http://homepages.warwick.ac.uk/staff/J.E.Cremona/courses/MA257/ma257.pdf

http://www2.math.uu.se/~astrombe/talteori2016/lindahl2002.pdf

43

Course Code

&Course Name M 303

Classical Mechanics

Objectives1. Beginning with a review of Newton's Laws applied to systems of particles.

2. The course moves on to rotational motion, dynamical gravity (Kepler's Laws) and motion in

non-inertial reference frames.

3. Students will know the concepts of classical mechanics and demonstrate a proficiency in the

fundamental concepts in this area of science.

4. Employ conceptual understanding to make predictions, and then approach the problem

mathematically. 5. Students will be able to solve problems using their knowledge and skills in modern physics.

SYLLABUS

Unit I

Learning

Out Comes

Introductory Ideas: Introduction, Space and Time, Newton’s Laws of Motion,

Inertial Frames, Gravitational Mass, Mechanics of a Particle: Conservation Laws,

Mechanics of a System of Particles, Lagrangian Dynamics: Introduction, Basic

Concepts, Constraints, Generalized Coordinates.

(Sections 1.1 to 1.7, 2.1 to 2.4 of the prescribed book.)

Having successfully completed this module, student will be able to

1. Understand the linear motion of systems of particles

2. Identify angular momentum for a particle and a system

3. Analyze, synthesize and process information.

Unit II

Learning

Out Comes

D’Alembert’s Principle, Lagrange’s Equations from D’Alembert’s principle,

Procedure for formation of Lagranges’s Equations, Lagrange’s Equations in

presence of Non-conservative forces, Generalized Potential – Lagrangian for a

Charged Practicle, Hamilton’s Priniciple and Lagrange’s Equations, Superiority of

Lagrangian Mechanics over Newtonian Approach, Guage Invariance of the

Lagrangian, Symmetry Properties of Space and Time and Conservation Laws,

Invariance under Galilean Transformation.

(Sections 2.6 to 2.14 of the prescribed text book.)

Having successfully completed this module, you will be able to

1. Describe and understand the motion of a mechanical system using Lagrange-

Hamilton formalism.

2. Analyze, synthesize and process information.

3. Utilize appropriate mathematical tools to analyze and solve a system’s equations

Unit III

Learning

Out Comes

Hamiltonian Dynamics: Introduction, Generalized momentum and cyclic

coordinates, Conservation Theorems, Hamiltonian Function H and Conservation of

Energy: Jacobi’s integral, Hamilton’s Equations, Hamilton’s Equations in different

coordinate systems, Examples in Hamiltonian Dynamics. (Sections 3.1 to 3.7 of

the prescribed text book.)


1. Discuss the linear motion of systems of particles (e.g. rocket motion)

2. Define angular momentum for a particle and a system

3. Define moment of inertia and use it in simple problems

Unit IV

Variational principles: Introduction, The Caclculus of Variations and Euler-

Lagranges’s Equations, Deduction of Hamilton’s Principle from D’Alembert’s

Principle, Modified Hamilton’s Principle, Deduction of Hamilton’s Equations

from modified Hamilton’s principle (Variational Principle), Deduction of

Lagrange’s Equations form Variational Principle for non-conservative systems,

44

Learning

Out Comes

Physical significance of Lagrange’s Multipliers λ, ∆-Variation, Principle of Least

Action, Other Forms of Principle of Least Action.

(Sections 5.1 to 5.8, 5.10 to 5.12 of the prescribed text book.)


1. Demonstrate knowledge and understanding of Lagrangian and Hamiltonian

formulation of mechanics.

2. Describe and understand the vibrations of discrete and continuous mechanical

systems.

3. Apply the Lagrangian formalism to analyze problems in Mechanics

Unit V

Learning

Out Comes

Dynamics of Rigid Body: Generalized Coordinates of a Rigid Body, Body and

Space Reference Systems, Euler’s Angles, Infinitesimal Rotations as Vectors-

Angular Velocity, Components of Angular Velocity, Angular Momentum and

Inertia Tensor, Principle Axes Principle Moments of Inertia, Rotational kinetic

Energy of a Rigid Body, Toque-Free Motion for a Rigid Body.

(Sections 10.1 to 10.12 of the prescribed text book.)


1. To demonstrate knowledge and understanding the dynamics of system of

particles,motion of rigid body.

2. Describe and understand planar and spatial motion of a rigid body.

3. Translate physical problems into appropriate mathematical language and apply

appropriate mathematical tools

Prescribed

Text Book

Reference

Books

Online

Source

Classical mechanics by J.C. Upadhyaya, Himalaya Publishing House Pvt. Ltd.

1. Classical mechanics by H. Goldstein, 2nd edition, Narosa Publishing House.

2. Classical Mechanics by Gupta, Kumar and Sharma

1. http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf

2. http://courses.physics.ucsd.edu/2010/Fall/physics200a/LECTURES/200_COURSE.pdf

Course

Outcomes

Upon successful completion, students will have the knowledge and skills to

1. Solve complicated physical problems using the principle of least action.

2. Describe the role of the wave equation and appreciate the universal nature of

wave motion in a range of physical systems.

3. Use Fourier theory and diffraction to describe properties of waves.

Understand the fundamentals of the mechanics of continuous systems.

4. Model and analyze the dynamics of physical systems using computational

methods.

5. Through the lab course, understand the principles of measurement and error

analysis and develop skills in experimental design.

http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf

http://courses.physics.ucsd.edu/2010/Fall/physics200a/LECTURES/200_COURSE.pdf

45

Course Code

&Course Name M304(1)

OPERATIONS RESEARCH

Objectives

1. To make the students familiar with principles and techniques of operations research and their

applications in decision making.

2.This course aims at familiarizing the students with quantitative tools and techniques, which are

frequently applied to business decision-making

3. To provide a formal quantitative approach to problem solving and an intuition about situations

where such an approach is appropriate.

4. To effective communicative ideas and explain procedures.

5. To effectively interpret results and solutions in written.

SYLLABUS

Unit I

Learning

Out Come

Linear Programming:DetermineThe Simplex Method – Overall Idea of the

Simplex Method – Development of the Simples Method – Primal Simplex method

– Dual Simplex Method – Special cases in Simplex Method Applications –

Sensitivity Analysis.


1.Explain the importance and scope of operations research

2. Formulate linear programming problems for resource allocation.

3. Solve linear programming problems using appropriate techniques.

Unit II

Learning

Out Come

Revised Simplex Method and Duality: Mathematical Foundations – Revised

(Primal) Simplex Method – Definition of the Dual Problem – Solution to the Dual

Problem – Economic Interpretation of the Dual Problem.


1. Solve linear programming problems using Revised Simplex method.

2. Solve linear programming problems using Dual Problem.

3. Applications of Economic Interpretation of the Dual Problem.

Unit III

Learning

Out Come

Determine the Transportation Model, Net Works and Applications of the

Transportation – Solution of the Transportation Problem – The Assignment Model

– The Transhipment Model.


1. Choose the method for initial solution of transportation problem.

2. Solve the transportation problem using optimum solution.

3. Solve the assignment problems using different methods.

Unit IV

Learning

Out Come

Network Definitions – Minimal Spanning Tree problem – Shortest – Route

Problem Network Models: - Maximal Flow Problem


1. Understand the concept of Network model

2. Applications of Shortest route network models

3. Allications of maximal flow models.

Unit V

Learning

Out Come

The Minimum Cost Capacitated Flow Problem Decision Theory and

Games:Decisions Under Uncertainty – Game Theory – Optimal solution of Two-

Person Zero-sum Games, Mixed strategies.


1. Understand the concept of Flow problems.

46

2. Applications of decision theory.

3. Applications of game theory.

Prescribed

Text Book

Reference

Books

Online

Source

1.Operations Research, An Introduction: Hamdy A Taha, Maxwell Macmillan

International Edition, New York, 1992.

1. Operations Research by Hira and Gupta, S.Chand Company and PVT LTd.

2. Operations Research by D N Mishra , S K Agarwal, PoojaSinha, World Press,

Luknow,U.P,India.

1. https://www.pdfdrive.com/operations-research-books.html

2. www.pondiuni.edu.in/storage/dde/downloads/mbaii_qt.pdf

3. https://easyengineering.net/operations-research-p-ramamurthy/

4. www.cs.toronto.edu/~stacho/public/IEOR4004-notes1.pdf

Course

Out Comes

At the end of the course students should be able to

1. Understand the mathematical tools that are needed to solve optimization.

2. Choose the method for effective allocation.

3. Interpret the relation between initial to optimum solution in transportation.

4. Evaluate the right person to the right job.

5. Choose the best strategy for business decision making.

https://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=D+N+Mishra&search-alias=stripbooks

https://www.amazon.in/s/ref=dp_byline_sr_book_2?ie=UTF8&field-author=S+K+Agarwal&search-alias=stripbooks

https://www.amazon.in/s/ref=dp_byline_sr_book_3?ie=UTF8&field-author=Pooja+Sinha&search-alias=stripbooks

https://www.pdfdrive.com/operations-research-books.html

https://www.pdfdrive.com/operations-research-books.html

http://www.pondiuni.edu.in/storage/dde/downloads/mbaii_qt.pdf

http://www.pondiuni.edu.in/storage/dde/downloads/mbaii_qt.pdf

https://easyengineering.net/operations-research-p-ramamurthy/

https://easyengineering.net/operations-research-p-ramamurthy/

47

Course Code


MATHEMATICAL BIOLOGY

Objectives

1.This course is aimed to be accessible both to Master's students of biology who have a good

understanding of the introductory course to mathematical biology and to Master's students in

Mathematics looking to broaden their application areas.

2. The course extends the range of usage of mathematical models in biology, ecology and

evolution.

3. Biologically, the course looks at models in evolution, population genetics and biological

invasions.

4. Mathematically the course involves the application of multivariable calculus, ordinary

differential equations and partial differential equations.

5. Formulation and analysis of ordinary differential equation (ODE) models for the population of

a single species, finding equilibrium populations and determining how their stability depends on

parameters.

SYLLABUS

Unit I

Learning

Out Comes

Autonomous differential equations - Equilibrium solutions - Stability nature of

equilibrium solutions, single species growth models involving exponential, logistic

and Gompertz growths. Harvest models – bifurcations and break points.

(Sections 1 and 2 of the Text Book1)

Upon completing this unit, students will be able to

1. Understand the concept of single species growth models involving exponential,

logistic and Gompertz growths. Harvest models – bifurcations and break points.

2. Convert verbal descriptions of biological systems into appropriate mathematical

models amenable to quantitative and qualitative analysis.

3. Develop the ability to explain mathematical results in language understandable

by biologists.

Unit II

Learning

Out Comes

LotkaVolterra predator – prey model – phase plane analysis, General predator prey

systems – equilibrium solutions – classification of equilibria – existence of cycles

– Bendixson-Dulac’s negative criterion – functional responses.

(Sections 7 and 8 of the text book1)


1. Understand the concept of LotkaVolterra predator – prey model – phase plane

analysis and applications of Bendixson-Dulac’s negative criterion.

2. Identify the equilibrium points and study the phase portrait analysis of predator

prey model.

3. Perform elementary mathematical analysis of models introduced and interpret

conditions obtained from the analysis - usually taking the form of relationships

between model parameters - that correspond to specific model behavior, and

express the ramifications for the biological process being considered.

Unit III

Learning

Out Comes

Global bifurcations in predator prey models – Freedman and Wolkowicz model -

type IV functional response – Hopf bifurcation – Homoclinic orbits – Global

bifurcations using Allee effect in prey – Competition models.

(Sections 9 and 10 of the prescribed text book1)


1. Understand the concept of Global bifurcations in predator prey models, Global

bifurcations using Allee effect in prey – Competition models and applications.

48

2. Understand and apply the concept of stability of a fixed point solution of a

system of ordinary differential equations.

3. Analyze the model with graphical representation and give biological

interpretation.

Unit IV

Learning

Out Comes

Lotka – Voltrrra Competition model – exploitation competition models. Mutualism

models – various types of mutualisms – cooperative systems – Harvest models and

optimal control theory (Sections 11 and 12 of the text book1)


1. Analyze ODE models for the populations of two interacting species.

2. Identify equilibrium points and using information about their linear stability to

characterize the long-term behavior of the system.


interpretation.

Unit V

Learning

Out Comes

Open access fishery – sole owner fishery – Pontryagin’s maximum principle –

Economic interpretation of Hamiltonian and adjoint variable.

(Sections 13 and 14 of the prescribed text book)


1. Understand the concepts of Open access fishery, sole owner fishery

2. Apply Pontryagin’s maximum principle to Open access fishery, sole owner

fishery

3. Analyze economical interpretation for sole owner fishery.

Prescribed

Text Book

Reference

Books

Online

Source

Elements of Mathematical Ecology by Mark Kot, Cambridge University Press,

2001.

1. Nisbet and Gurney, 1982, Modelling Fluctuating Populations, John Wiley &

Sons.

2. Modeling through Differential Equation by D. N. Burghes Ellis Horwood and

John Wiley.

3. Principle of Mathematical Modeling by C. Dyson and E. Levery, Academic

Press New York.

4. A First Course in Mathematical Modeling by Giordano, Weir, Fox 2nd Edition,

Brooks/ Cole Publishing Company, 1997.

5. Mathematical Modeling by J. N. Kapur, Wiley Eastern Ltd. 1994.

6. Mathematical Modeling with Case Studies B. Barnes, G. R. Fulford, A

Differential Equation Approach using Maple and Matlab, 2nd Ed., Taylor and

Francis group, London and New York, 2009.

1. https://www.math.mun.ca/~zhao/ARRMSschool/MathBioNotes2011.pdf

2. http://www.di.univr.it/documenti/OccorrenzaIns/matdid/matdid262230.pdf

Course Out

Comes

On successful completion of this course unit students will be able to

1. Read, situate, and understand research papers in the area of mathematical

biology.

2. Prepare to discuss specific biological systems with life scientists, and in

particular communicate efficiently how values of model parameters can impact the

qualitative behavior of the system.

3. Solve mathematically and interpret biologically simple problems involving one-

and two-species ecosystems, epidemics and biochemical reactions.


interpretation for competition models, mutualism models.

5. Analyze economical interpretation for open access fishery, sole owner fishery

models using Pontryagin’s maximum principle.

https://www.math.mun.ca/~zhao/ARRMSschool/MathBioNotes2011.pdf

http://www.di.univr.it/documenti/OccorrenzaIns/matdid/matdid262230.pdf

49

Course Code


Commutative Algebra I

Objectives

1.The course develops the theory of commutative rings.

2.These rings are of fundamental significance since geometric.

3.Number theoretic ideas is described algebraically by commutative rings

4.Knows basic definitions concerning elements in rings, classes of rings, and ideals in

commutative rings.

5.Can use algebraic tools which are important for many problems and much theory development

in algebra.

SYLLABUS

Unit I

Learning

Out Comes

Rings and ring homomorphism, ideals, quotient rings, zero divisors, Nilpotent

elements,units, prime ideals and Maximal ideals.


1. Understand the concept of Ring theory.

2. Understand the concept of different idels.

3. Applications of Ring theory.

Unit II

Learning

Out Comes

Nil radical and Jacobson radical, operations on ideals, Extensions and

contractions-Modules and module homomorphisms, Sub modules and quotient

modules, operations onsubmodules, Direct sum and product, finitely generated

modules.


1. Express the concept of radical and Jacobson radical.

2. Understand the concept of Modules and its applications.

3. Understand the concept of Direct sum and product.

Unit III

Learning

Out Comes

Exact sequences, Tensor product of modules, Restriction and extension of

scalars,Exactness properties of the tensor product, algebras, tensor product of

algebras.


1. Understand the concept of Exact sequence, Tensor product.

2. Express the properties of Exactness and algebra.

3. Understand the concept of tensor product of algebras.

Unit IV

Learning

Out Comes

Local Properties- Extended and contracted ideals in rings of fractions.


1. Understand the concept of Local properties.

2. Application of the concept of Local properties.

3. Understand the Extended &contracted ideals in rings.

Unit V

Learning

Out Comes

The Primary decompositions.


1. Understand the concept of Decomposition.

2. Understand the concept of Primary Decomposition.

3. Application of Primary Decomposition.

Prescribed

Text Book

Introduction to commutative algebra, By M.F. ATIYAH and I.G.

MACDONALD, Addison-Wesley publishing Company, London.

50

Reference

Books

Online

Source

1. Basic Commutative Algebra by Balwant Singh, World scientific Publishing Co.

Pte. Ltd.

2. Commutative Algebra by N.S.Gopal Krishna, Second Edition.

1. web.mit.edu/18.705/www/13Ed.pdf

2. math.uga.edu/~pete/integral.pdf

3.https://www.jmilne.org/math/xnotes/CA.pdf

4. www.math.toronto.edu/jcarlson/A--M.pdf

Course

Out Comes

After studying this course, you should be able to

1. Knows basic definitions concerning elements in rings, classes of rings, and

ideals in commutative rings.

2. Know constructions like tensor product and localization, and the basic theory for

this.

3. Know basic theory for noetherian rings and Hilbert basis theorem.

4. Know basic theory for integral dependence, and the Noether normalization

lemma.

5. Have insight in the correspondence between ideals in polynomial rings, and the

corresponding geometric objects: affine varieties.

http://web.mit.edu/18.705/www/13Ed.pdf%202.%20math.uga.edu/~pete/integral.pdf%203.https:/www.jmilne.org/math/xnotes/CA.pdf%204.%20www.math.toronto.edu/jcarlson/A--M.pdf




51

Course Code


Banach Algebras

Objectives: 1. Banach algebras have a lot of structure, combining the topological

features of a Banach space with the algebraic features of a ring.

2. Although we shall see many other examples, our main focus will be onexaminingBanach

algebras consisting of continuous linear operatorsonHilbert and Banach spaces.

3. Students will able to learn the structure of commutative Banach Algebras.

4. Knows basic definitions concerning elements in rings, classes of rings, and ideals in

commutative rings.

5. Can use algebraic tools which are important for many problems and much theory development

in banachalgebra.

SYLLABUS

Unit I

Learning

Out Come

General preliminaries on Banach Algebras – The definition and examples –

Regular and singular elements – Topological divisors of Zero – The spectrum


1. Understand the concept of General preliminaries on Banach Algebras.

2. Application of the concept of Local properties and Regular and singular

elements.

3. Understand the Topological divisors of Zero.

Unit II

Learning

Out Come

The formula for the spectral radius – The radical and the semi – simplicity.


1. Understand the concept of Banach Algebras.

2. Application of the concept of formula for the spectral radius

3. Understand the radical and the sem-simplicity.

Unit III

Learning

Out Come

The structure of commutative Banach Algebras - TheGelfand mapping -

Applications of the formula r(x) = lim // xn // 1/n – Involutions in Banach Algebras

– The Gelfand – Neumark theorem.


1. Understand the concept of commutative Banach Algebras.

2. Application of the concept of TheGelfand mapping.

3. Understand the Involutions in Banach Algebras.

Unit IV

Learning

Out Come

Some special commutative Banach Algebras - Ideals in C(x) and the Banach –

Stone theorem - The stone – Cechcompactification – commutaticve C* - algebras.


1. Understand the concept of Ideals in C(x) and the Banach – Stone theorem.

2. Application of the concept of The stone – Cechcompactification.

3. Understand the commutaticve C* - algebras.

Unit V

Learning

Out Come

Fixed point theorems and some applications to analysis – Brouwer’s and

Schauder’s fixed point theorems (without proofs) Picard’s theorem – Continuous

curves – The Hahn – Mazurkiewicz theorem (without proof). Boolean rings – The

stone representation theorem.


1. Understand the concept of Fixed point theorems and some applications to

analysis.

52

2. Application of the concept of Picard’s theorem – Continuous curves.

3. Understand The Hahn – Mazurkiewicz theorem.

Prescribed

Text Book

Reference

Books

Online

Source

1.Introduction to Topology and Modern Analysis – By G.F. Simmons –

International Student edition – McGraw – Hill Kogakusha Ltd.

1. W.B. Arveson, A Short Course in Spectral Theory. (Chapters 1 and 2).

2. J.B. Conway, A Course in Functional Analysis. (Chapters 7, 8 and 9).

3.P.R. Halmos, A Hilbert Space Problem Book, (2nd ed.), Springer-Verlag, 1982.

1. http://www.math.nagoya-u.ac.jp/~richard/teaching/s2014/Course_Wilde.pdf 2. http://www.math.lmu.de/~petrakis/INTRODUCTION%20TO%20BANACH%20ALGEBRAS.pdf

3. https://www.math.uni-hamburg.de/home/khomskii/papers/Bachelor_Thesis_Yurii_Khomskii.pdf

Course

Out Comes

After completion of this course, student will be able to

1. Students taking this course will develop an appreciation of the basic concepts of

Functional Analysis, including the study of operator theory and the study of

topological function spaces.

2.These methods will be useful for further study in arange of other fields, e.g.

Quantum Theory, Stochastic calculus and Harmonicanalysis.

3. Use advanced theoretical and practical knowledge gained in the field.

4. Play a role in decision-making process based on the arising problems while

working with different disciplines.

53

Course Code

&Course Name M 305(1)

NUMERICAL ANALYSIS

Objectives

1.This course is aimed to be accessible both to Master's students of Mathematics who have a good

understanding of the introductory course to Numerical Analysis and to Master's students in

Mathematics looking to broaden their application areas.

2. The course extends the range of several available solutions of equations in one and more

variable.

3. Formulation and analysis of the several available methods to Solve the simultaneous equations.

4. Mathematically the course involves of Single step methods, multi step Method and obtaining

numerical solutions to problems of mathematics.

5. Analyze and evaluate numerical methods for various mathematical operations and tasks, such

as interpolation, differentiation, integration, the solution of linear and nonlinear equations, and the

solution of differential equations.

SYLLABUS

Unit I

Learning

Out Comes

Interpolation and Approximation: Introduction-Lagrange and Newton

interpolation-Finite difference operators- Interpolating polynomials using finite

differences – Hermite Interpolations.

Section 4.1 to 4.5 of Chapter-IV of the prescribed text book 1.


1. Construct a function which closely fits given n-points in the plane by using

interpolation method.,

2. Find the lagrange polynomial passing through the given points.

3. Find the hermite polynomial passing through the given points.

4. Find the cubic spline passing through the given points.

Unit II

Learning

Out Comes

Differentiations and Integration: Introduction – Numerical differentiation –

optimum choice of step length – extrapolation methods.

Section 5.1 to 5.4 of Chapter V of the prescribed text book 1.


1. Describing and understanding of the several errors and approximation in

numerical methods.

2. Find the optimum choice of step length problems

3. Demonstrate understanding of common numerical extrapolation methods.

Unit III

Learning

Out Comes

Numerical Integration – Methods based on interpolation – method based on

undetermined coefficients – composite Integration methods - Romberg Integration.

Section 5.6 to 5.10 of Chapter V of the prescribed text book 1.


1. Express the method based interpolation

2. Understanding of several available solutions of equations in one and more

variable.

3. Find the solution of an equation by the composite Integration methods -

Romberg Integration.

Unit IV

Learning

Ordinary differential Equations: Introduction – Numerical Method – single step

methods( Taylor series Method)

Section 6.1 to 6.3 of Chapter VI of the prescribed text book 1.


54

Out Comes 1. Analyse and evaluate the accuracy of common numerical methods.

2. Find the determined function using Taylor series Method.

3. Find the solution of an equation by the single step methods.

Unit V

Learning

Out Comes

Ordinary differential Equations:Single step methods (Runge - Kutta2nd order

and 4th order Methods, Estimation of Local truncation error, system of Equations)

– Multi step Method.

Section 6.3 to 6.4 of Chapter VI of the prescribed text book 1.


1. Understand thefundamental concepts of Single step methods and multi

stepMethod

2. Find the solution of an equation by the Runge – Kutta2nd order and 4th order

Methods.

3. Derive numerical methods for various mathematical operations and tasks such

as interpolation, difference of linear and nonlinear equations and the solutions of

differential equations.

Prescribed

Text Book

Reference

Books

Online

Source

1.Numerical Methods, by M.K. Jain, S.R.K.Iyengar, R.K. Jain, 3rd Edition

New Age International (P) Limited, Publishers.

1.Numerical Methods, by M.K. Jain, S.R.K.Iyengar, R.K. Jain, 6th Edition

New Age International (P) Limited, Publishers.

2. Introductory Methods of Numerical Analysis by S.S. Sastry, 4th edition PHI

Publication.

3. K. E. Atkinson, An Introduction to Numerical Analysis (2nd edition), Wiley-

India, 1989.

4. S. D. Conte and Carl de Boor, Elementary Numerical Analysis - An Algorithmic

Approach (3rd edition), McGraw-Hill, 1981.

1. https://www.math.ust.hk/~machas/numerical-methods.pdf

2. http://www.math.iitb.ac.in/~baskar/book.pdf

Course

Out Comes


1. Have advanced theoretical and practical knowledge to comprehend textbooks,

scientific papers etc. containing current information

2. Use advanced theoretical and practical knowledge gained in the field.

3. Update the theoretical and practical knowledge depending on the current

conditions.

4. Play a role in decision-making process based on the arising problems while

working with different disciplines.

https://www.math.ust.hk/~machas/numerical-methods.pdf

http://www.math.iitb.ac.in/~baskar/book.pdf

55

Course Code

&Course Name M 305 (2)

Mathematical Software

Objectives

1. This course provides mathematical software to write mathematical notes, projects, and articles,

solving equations and plotting functions of one, two and three variables.

2. Student will explore: Latex, Matlab for numerical computation and Maple

3. To enable the student on how to approach for solving Engineering problems using simulation

tools.

4. Numeric and symbolic tools for discrete and continuous calculus including definite

and indefinite integration, definite and indefinite summation, automatic differentiation and

continuous and discrete integral transforms

5. Plotting of function of one, two and three variables using maple.

SYLLABUS

Unit I

Learning

Out Comes

LATeX introduction- Installation – Math symbols and tables – TeX symbol and

tables – Matrix and lists – Typing Math and text – Text environments.


1. Know about latex software.

2.Learn how to write math symbols in latex software

3. Write list of mathematical symbols and some paragraph.

Unit II

Learning

Out Comes

Document structure – Latex Documents – The AMS articles document class –

Bemer Presentation and PDF documents – Long Documents – BibteX – Make

index – Books in LateX- Colours and Graphics – TeXCAD – LATeX CAD.


1. Know about document structure, bemer presentation.

2. Learn how to write Document structure – Latex Documents – The AMS articles

document class.

3. Write books in latex – colours and graphics.

Unit III

Learning

Out Comes

Starting with MATLAB- Variables Vectors, Matrices – Creating Array in

MATLAB –Menu, Workspace, working Directory, Command window, Diary,

Printing- Built-in function, User defined functions, Script M-files- Complex

Arithmetic, Eigen values and Eigen vectors – Two and three dimensional Plots.


1. Know about MATLAB software.

2. Learn Variables Vectors, Matrices – Creating Array in MATLAB.

3. Create M-files and Plot a data diagram.

Unit IV

Learning

Out Comes

Getting around with maple – Maple input and output - Programming in Maple.


1. Know about maple software.

2. Learn input and output data, programming in maple.

3. Plot a data diagram.

Unit V

Learning

Maple: Abstract algebra – Linear algebra – Calculus on numbers – Variables-

Complex Arithmetic, Eigen values and Eigen vectors – Two and three dimensional

plots.


https://en.wikipedia.org/wiki/Calculus

https://en.wikipedia.org/wiki/Indefinite_integration

https://en.wikipedia.org/wiki/Indefinite_sum

https://en.wikipedia.org/wiki/Integral_transform

56

Out Comes 1. Know about maple software and write maple code.

2. Calculate arithmetic expressions, eigenvalues and eigenvectors.

3. Plot a data diagram.

Prescribed

Text Book

Reference

Books

Online

Source

1. G. Gratzer, More Math Into LATEX, 4th edition, Springer, (2007).

2. AMOS Gilat, MATLAB an introduction with application, WILEY India

Edition, (2009).

3. Brain R Hunt, Ronald L Lipsman,A Guide to MATLAB for beginners and

Experienced users, Cambridge University Press. (2003)

4. Ander Heck, Introduction in Maple, Springer, (2007)

1. LaTeX Beginner's Guide Kindle Editionby Stefan Kottwitz

2. MATLAB: A Practical Introduction to Programming and Problem Solving by

Stormy Attaway.

3. Understanding maple by Ian Thompson.

1. http://www.docs.is.ed.ac.uk/skills/documents/3722/3722-2014.pdf

1.http://page.math.tu-berlin.de/~chern/notes/MatlabLectureNote.pdf

2.http://mayankagr.in/images/matlab_tutorial.pdf

4. https://www.eecs.umich.edu/dco/docs/maple/intropg.pdf

Course Out

Comes

After completing this course students will be able to

1. Gain knowledge of Latex software and learn to write mathematical symbols in

Latex.

2. Write mathematical projects in Latex software and prepare mathematics notes.

3. Know uses of Matlab software and solve some mathematical expressions using

codes in Matlab.

4. Plot diagrams by writing codes in Matlab.

5. Solve for systems of equations, ODEs, PDEs and recurrence relations writing

codes in maple.

https://www.amazon.in/s/ref=dp_byline_sr_ebooks_1?ie=UTF8&text=Stefan+Kottwitz&search-alias=digital-text&field-author=Stefan+Kottwitz&sort=relevancerank

https://www.amazon.in/s/ref=dp_byline_sr_book_1?ie=UTF8&field-author=Stormy+Attaway+Ph.D.++Boston+University+Dr.&search-alias=stripbooks

http://www.docs.is.ed.ac.uk/skills/documents/3722/3722-2014.pdf

http://page.math.tu-berlin.de/~chern/notes/MatlabLectureNote.pdf

http://mayankagr.in/images/matlab_tutorial.pdf

https://www.eecs.umich.edu/dco/docs/maple/intropg.pdf

https://en.wikipedia.org/wiki/Systems_of_equations

https://en.wikipedia.org/wiki/Ordinary_differential_equation

https://en.wikipedia.org/wiki/Partial_differential_equation

https://en.wikipedia.org/wiki/Recurrence_relation

57

Course Code


Fuzzy Set Theory

Objectives

1. Provide an understanding of the basic mathematical elements of the

theory of fuzzy sets.

2. Provide an emphasis on the differences andsimilarities between fuzzy

sets and classical sets theories.

3. The mainobjective of this course is to establish thorough background

knowledgeon evolutionary algorithms in post graduate students.

4. The students topursue individual research in solving real world

optimization problemslike Constrained, Multimodal, Multi objective and

CombinatorialOptimizations.

5. Students will explore about Combinations of operations - Aggregation

Operations

SYLLABUS

Unit I

Learning

Out Comes

From Classical(Crisp) sets to fuzzy sets:- Introduction-Crispsets: An overview-

fuzzyset:Basic types-Fuzzy sets. Basic Concepts-Characteristics and significance

of the paradigm shift (Chapter-1).

Fuzzysets versus Crisp sets-Additional Properties of a𝛼-cuts-Representations of

Fuzzysets-Extension principle for Fuzzysets (Chapter-2).

After studying this unit, student will able to

1. Learn Basic types-Fuzzy sets

2. Know about Fuzzysets versus Crisp sets-Additional Properties of a cuts-

Representations of Fuzzysets

Unit II

Learning

Out Comes

Operations on Fuzzysets - Types of Operations - Fuzzy Compliments - Fuzzy

Inter sections: t-norms - Fuzzy unions; t-Conorms - Combinations of operations -

Agreegation Operations (Chapter-3).


1. Know aboutFuzzy Compliments - Fuzzy Inter sections

2. LearnFuzzy unions; t-Conorms

3. Know about Combinations of operations - Agreegation Operations

Unit III

Learning

Out Comes

Fuzzy Arithmetic -Fuzzy Numbers - Linguistic variables - Arithmetic operations

on intervals - Arithmetic operations on Fuzzy numbers - Lattice of fuzzy numbers -

Fuzzy equations (Chapter-4).


1. Know about Fuzzy Numbers - Linguistic variables

2. Know about Arithmetic operations on intervals

3. Learn Lattice of fuzzy numbers - Fuzzy equations

Unit IV

Learning

Out Comes

Fuzzy Relations - Crisp versus fuzzy relations - Projections and Cylindric

Extensions - Binary Fuzzy Relations - Binary Relations and Singleset - Fuzzy

Equivalence Relations. (5.1-5.5 in Chapter-5)


1. Know about Crisp versus fuzzy relations

58

2. Know about Projections and Cylindric Extensions

3. LearnBinary Relations and Singleset - Fuzzy Equivalence Relations.

Unit V

Learning

Out Comes

Binary Relations on a single set - Fuzzy Compatibility Relations - Fuzzy Ordering

Relations - Fuzzy Morphisms - Sup - Compositions of Fuzzy Relations - Inf -

Compositions of fuzzy Relations. (5.6-5.10 in Chapter-5)


1. Learn about Binary Relations on a single set.

2. Know about Fuzzy Morphisms - Sup - Compositions of Fuzzy Relations.

3. Express Inf - Compositions of fuzzy Relations

Prescribed

Text Book

Reference

Books

Online

Source

G.J.KLIR and BOYUAN "Fuzzy sets and Fuzzy Logic, Theory and Applications"

Prentice - Hall of India Pvt. Ltd., New Delhi., 2001.

1. H.J. Zimmermann, “Fuzzy set theory and its Applications “Allied Publishers

Ltd., New Delhi,1991 (For Units I & II).

2. Yu, Xinjie, Gen, Mitsuo, “Introduction to Evolutionary Algorithms”, Spinger,

ISBN 978- 1-84996-129-5.

1. T.J. Ross, John Wiley & Sons, Fuzzy Logic with Engineering Applications”,

IInd Ed., 2005.

2. M.C. Bhuvaneswari, “Application of Evolutionary Algorithms for Multi-

objectiveOptimization in VLSI and Embedded Systems”, Spinger,2014.

3. Ashlock, D. (2006), “Evolutionary Computation for Modeling and

Optimization”, Springer, ISBN0-387-22196-4.

1. https://www.mv.helsinki.fi/home/niskanen/zimmermann_review.pdf

2. https://cours.etsmtl.ca/sys843/REFS/Books/ZimmermannFuzzySetTheory2001.pdf

3. http://logica.dipmat.unisa.it/lucaspada/wp-content/uploads/foligno_handout.pdf

Course Out

Comes

1. Students taking this course will develop an appreciation of the basic concepts of

Functional Analysis, including the study of operator theory and the study of

topological function spaces.

2. These methods will be useful for further study in a range of other fields, e.g.

Quantum Theory, Stochastic calculus and Harmonic analysis.

3. Awareness of advanced theoretical and applied information supported by the

statistical sources.

4. Awareness of the significance and impact of statistical methods on the social

dimensions of interdisciplinary studies.

5. The ability to share solutions related with the problems encountered in the field

with experts and non-experts by supporting quantitative and qualitative data.

https://www.mv.helsinki.fi/home/niskanen/zimmermann_review.pdf

https://cours.etsmtl.ca/sys843/REFS/Books/ZimmermannFuzzySetTheory2001.pdf

59

Course Code


Universal Algebra

Objectives:1.One of the aims of universal algebra is to extract the common elements of

seemingly different types of algebraic structures such as groups, rings or lattices.

2.Doing so one discovers general concepts, constructions, and results which unify and generalize

the known special situations.

3.Applications of universal algebra can be found in logic through the interface of algebraic logic.

4.The course will introduce the students to the basic concepts and theory of universal algebra.

SYLLABUS

Unit I

Learning

Out Comes

Definitions of Lattices – Isomorphisms of Lattices and Sub lattices- Distributive

andModular Lattices- Complete lattices- Equivalence relations- Algebraic lattices.

Sections 1 to 4 of Chapter 1 of the prescribed text book.

After the completion of the unit, Students will be able to

1. Distinguish isomorphism of lattices and sub lattices.

2. Classify the modular and distributive lattices.

3. Learn applications equivalence relations.

Unit II

Learning

Out Comes

Closure operators, Definition and examples of algebras- Isomorphic algebras and

subalgebras – Algebraic lattices and sub universes.

Section 5 of Chapter 1 & Sections 1 to 3 of Chapter 2 of the prescribed text book.


1.Classify Algebras-lattices- closure operators.

2. Determine the isomorphic algebras and sub algebras.

3. Understand the concept of algebraic lattices and sub universes.

Unit III

Learning

Out Comes

The irredundant Basis theorem- Congruences and Quotient algebras.

Homomorphisms –The homomorphism and isomorphism theorems, Direct

products- Factor congruences –Directly indecomposable algebras.



1. Determine the concept of irredundant basis theorems.

2. Understand the concept of homomorphism and isomorphism theorems.

3. Understand the concept of directly indecomposable algebras.

Unit IV

Learning

Out Comes

Sub direct products- Subdirectly irreducible algebras- Simple algebras- Class

operators-Varieties.Terms- Term algebras- Free algebras.


1.Understand the concept of sub direct products.

2.Determine the concept of subdirectly irreducible algebras , simple algebras.

3. Understand the concept of terms ,term algebras ,free algebras.

60

Unit V

Learning

Out Comes

Identities and Free algebras- Birkhoff’s theorem- Malcev conditions- The Centre

of analgebra.


1. Understand the concept of Ideals and dual ideals.

2. Determine the concept of Ideal chains- Ideal lattices

3. Understand the concept of Distributive lattices and rings of sets.

Prescribed

Text Book

Reference

Books

Online

Source

A course in Universal algebra- Stanley Burris, H.P. Sankappanavar,Springer-

Verlag, New York- Heidelberg- Berlin.

1. A Course in. Universal Algebra. H. P. Sankappanavar. Stanley Burris.

2. Universal Algebra: Fundamentals and Selected Topics Book by Clifford

Bergman.

3. Universal Algebra Book by Paul Cohn.

1. https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf

2. https://link.springer.com/book/10.1007/978-0-387-77487-9 3.https://kluedo.ub.unikl.de/frontdoor/deliver/index/docId/1493/file/universal_algebra.pdf

Course Out

Comes

1.Recognise technical terms and appreciate some of the uses of algebra

2.Collect like terms and simplify expressions term by term

3.Multiply out brackets

4.Simplify some formulas

5.Solve simple linear equations.

https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf

61

M.Sc Mathematics SECOND YEAR – FORTH SEMESTER

Course Code &

Course Name M 401

MEASURE & INTEGRATION

Objectives

1. In this course we will learn the basic concepts of Measure theory and integration theory, with

related discussions on applications in classical Banach spaces.

2. The solution methods studied in this course will include the basic curriculum since it is crucial

for understanding the theoretical basis of probability and statistics.

3. Introduce students to how to prove bounded convergence theorem, Fatcous lemma and

monotone convergence theorem.

4. The course helps students develop skills to think quantitatively and analyse problems in absolute

continuity, convex functions and Bounded variation functions

5. This course will be useful for majors in such as harmonic analysis, ergodic theory, theory of

partial differential equations and probability theory.

SYLLABUS

Unit I

Learning

outcomes

Lebesgue measure: Introduction, Outer measure, measurable sets and Lebesgue

measure, A non-measurable set, measurable functions, Littlewood’s three

principles.

Chapter 3 of the text book

After the unit the students are expected to be able to:

1. Elaborate how to demonstrate a property for measurable sets and Lebesgue measure

2. Demonstrate understanding of non-measurable sets

3.Express the Littlewood’s three principles

Unit II

Learning

outcomes

The Lebesgue Integral: The Riemann integral, The Lebesgue integral of a

bounded function over a set of finite measure, the integral of nonnegative function.


By the end of the unit, the students must be able to:

1. Define and understand basic notations in riemann integral.

2.Determine conditions for the convergence of integrals

3.Apply integral convergence theorem to obtain approximate solutions to

mathematical problems

Unit III

Learning

outcomes

The Lebesgue Integral:The general Lebesgue integral, convergences in measure,

Differentiation and integration: Differentiation of monotone functions,


By the end of the unit, the students will be able to:

1.Describe the notion of convergences in measure

62

2.Establish basic properties for monotone functions,

3. Prove the bounded convergence theorem, Fatcous lemma and monotone

convergence theorem.

Unit IV

Learning

outcomes

Differentiation and integration:Functions of bounded variation and

differentiation of an integral, Absolute continuity, and convex functions.

Chapter 5 and Chapter 6 of the text book

1. Acquire knowledge of bounded variation functions

2. Define and understand the concept of Absolute continuity, and convex

functions.

3.Learn these apply for real world problem.

Unit V

Learning

outcomes

The classical Banach spaces: The Lp-spaces, The Minkoswki and Holder

inequalities, convergence and completeness, approximation in Lp, Bounded linear

functionals on the Lp spaces.


1. Understand the definition of Lp spaces.

2. Apply holders and minkowskis inequalities.

3. describeRiesz representation theorem.

Prescribed

Text Book

Reference

Books

Online

Source

1. Real Analysis by H. L. Royden, Macmillan Publishing Co. Inc. 3rd Edition, New

York, 1988.

1.Jones, Frank. Lebesgue Integration on Euclidean Space. Boston: Jones & Bartlett

Publishers, February 1, 1993.

2.Evans, Lawrence C., and Ronald F. Gariepy. Measure Theory and Fine

Properties of Function. Boca Raton, Florida: CRC Press, December 18, 1991.

ISBN: 0849371570.

1. https://people.math.ethz.ch/~salamon/PREPRINTS/measure.pdf.

2. http://www.math.utoronto.ca/almut/MAT1000/LL-1.pdf.

3. http://www.math.nagoya-u.ac.jp/~richard/teaching/s2017/Nelson_2015.pdf.

Course

Out Comes


1. Computation of Lebesgue measures.

2. Establishing measurability or non-measurability of sets and functions. 3.

Approximating measurable functions by simple and step functions.

4. Computation of Lebesgue integrals, applications to volume calculations and

Fourier analysis.

5. Deciding under which conditions the fundamental theorem of calculus is

applicable in the context of Lebesgue integration.

63

Course Code

&Course Name M 402

PARTIAL DIFFERENTIAL EQUATIONS

Objectives 1. In this course we will study first and second order partial differential equations.

2. The solution methods studied in this course will include the method of characteristics,

separation of variables.

3. Introduce students to how to solve linear Partial Differential with different methods.

4. Technique of separation of variables to solve PDEs and analyze the behavior of solutions in

terms of eigenfunction expansions.

5. This course will be useful for majors in economics, mathematical finance, engineering and

physics.

SYLLABUS

Unit I

Learning

Out Comes

First Order P.D.E.: Curves and Surfaces – Genesis of First Order P.D.E. –

Classification of Integrals – Linear equations of the First Order - Pfaffian

Differential Equations – Compatible Systems.

Chapter 1 – sections 1.1 – 1.6


1. Distinguish ordinary differential equations and partial differential equation and

understand the concepts of first order -Pfaffian differential equations – compatible

Systems.

2. Classify the first order partial differential equations.

3. Learn applications of First order partial differential equations.

Unit II

Learning

Out Comes

Charpit’sMethod Jacobi’s Method - Integral Surfaces Through a Given Curve–

Quasi Linear Equations.

Chapter 1 – sections 1.7 – 1.11


1. Use the method of undetermined coefficients to solve second order, linear

homogeneous equations with constant coefficients

2. Use the method of reduction of order to find a second linearly independent

solution of a second order, linear homogeneous equation when one solution is

given.

3. Understand the notion of linear independence and the notion of a fundamental

set of solutions

Unit III

Second Order P.D.E.: Genesis of Second Order P.D.E. – Classification of Second

Order P.D.E. – One Dimensional Wave equation: Vibrations of an Infinite String –

Vibrations of Semi infinite String Vibrations of a String of Finite Length –

Riemann’s Method – Vibrations of a String of Finite Length (Method of

Seperation of Variables).

64

Learning

Out Comes

Chapter 2 – section 2.1 – 2.3.


1. Classify partial differential equations and transform into canonical form.

2. Use the method of variation of parameters to find particular solutions of second

order, linear homogeneous equations

3. Analyzestring of finite length, string of infinite length.

Unit IV

Learning

Out Comes

Laplace’s Equation: Boundary value Problems- Maximum and Minimum

Principles- The Cauchy Problem – The Dirichlet Problem for the Upper Half Plane

– The Neumann Problem for the Upper Half Plane – Dirichlet Problem for a Circle

– The Dirichlet Exterior Problem for a Circle- The Neumann Problem for a Circle

– The Dirichlet Problem for a Rectangle- Harnack’s Theorem Laplace’s Equation

– Green ‘s Function – The Dirichlet Problem for a Half Plane – The Dirichlet

problem for a Circle.

Chapter 2 – section 2.4


1. Understand the concepts of boundary value problems,maximum and minimum

principles, the Cauchy Problem.

2. Construct the Green’s function for Partial differential equations.

3. Apply Neumann problem for the upper half plane, for a circle.

Unit V

Learning

Out Comes

Heat Conduction Problem: Heat Conduction – Infinite Rod Case - Heat

Conduction –Finite Rod Case- Duhamel’s Principle –Wave Equation –Heat

Conduction Equation

Chapter 2 – sections 2.5, 2.6.


1. Understanding the heat conduction equation.

2. Apply Duhamel’s Principle for wave equation, heat conduction equation.

3. Apply partial derivative equation techniques to predict the behaviour of certain

phenomena.

Prescribed

Text Book

Reference

Books

Online

Source

An Elementary Course in Partial differential equations by T. Amarnath, Second

Edition, Narosa Publishing House, 1997.

1. Elements of Partial Differential Equations by Ian Sneddon, International

Students Edition.

2. Partial Differential Equations by Phoolan Prasad and RenukaRavindran, New

Age International, 1985.

3. Partial Differential Equations by F. John, Springer-Verlag, New York, 1978. 20

4. Partial Differential Equations by Tyn-Myint-U, North Holland Publication, New

York, 1987.

5.Partial Differential Equations for Engineers and Scienistsby J. N. Sharma, K.

Singh, Narosa, 2nd Edition.

1. http://www.math.tifr.res.in/~publ/ln/tifr70.pdf

2. https://www.iist.ac.in/sites/default/files/people/PDE-Notes2.pdf

3. https://www.math.uni-leipzig.de/~miersemann/pdebook.pdf

Course

Out Comes

Upon successful completion of the course, students will have the knowledge and

skills to

1. Apply a range of techniques to find solutions of standard Partial Differential

Equations (PDE).

2. Understand basic properties of standard PDE's.

3. Demonstrate accurate and efficient use of Fourier analysis techniques and their

http://www.math.tifr.res.in/~publ/ln/tifr70.pdf

https://www.iist.ac.in/sites/default/files/people/PDE-Notes2.pdf

https://www.math.uni-leipzig.de/~miersemann/pdebook.pdf

65

applications in the theory of PDE's.

4. Demonstrate capacity to model physical phenomena using PDE's (in particular

using the heat and wave equations).

5. Apply specific methodologies, techniques and resources to conduct research and

produce innovative results in the area of specialisation.

Course Code

&Course Name M 403

MATHEMATICAL METHODS

Objectives

1. This course aims to develop a basic understanding of a range of mathematics tools with

emphasis on engineering applications.

2. It is intended for students to solve problems with techniques from advanced linear algebra,

ordinary differential equations and multi-variable differentiation.

3. Laplace transforms are also introduced.

4. The course helps students develop skills to think quantitatively and analyse problems critically.

5. This course addresses a number of important mathematical methods often used in physics.

SYLLABUS

Unit I

Learning

Out Comes

Laplace Transforms: Introduction – a few Remarks on the theory – Application to

differential equation – derivatives and intigrals of Laplace Transforms –

convolutions and Abel’s mechanical problem.

Sections 50 to 54 of Chapter 10 of the prescribed text book I.

On completion of this module, the learner will be able to

1. Describe several areas of Laplace transformation and applications.

2. Solving and model applied problems.

3.Determinederivatives and intigrals of Laplace Transforms and Abel’s mechanical

problem.

Unit II

Learning

Out Comes

Volterra Integral Equations: Basic concepts – Relationship between linear

differential equation and volterra Integral Equations – Resolvent kernel of Volterra

Integral Equations. Solution of Integral Equations by Resolvent kernel – The

Method of successive approximations – convolution type equation – solution of

integro- differential equation with the aid of the Laplace transformation

Sections 1 to 6 of Chapter I of the prescribed text book II.


1. Describe several areas of Integral Equations

2.Determine the Method of successive approximations and convolution type

equation

3.Understanding the concept of Integro- differential equation with the aid of the

Laplace transformation

Unit III

Fredholm integral equations: Fredholm integral equations of 2nd kind.

Fundamentals – the method of Fredholm determinants – iterated kernals.

Constructing the resolvent kernel with the aid of iterated kernals- integral

equations with degenatekernals. Hammerstein type equation- characterstic

numbers and eigen functions.

Sections 12 to 16 of Chapter II of the prescribed text book II.

66

Learning

Out Comes


1. Understanding the concept Fredholm integral equations

2. Describe the concept of resolvent kernel with the aid of iterated kernals

3. Solving the applications of characterstic numbers and eigen functions.

Unit IV

Learning

Out Comes

Solution of homogeneous of integral equations with degenate kernel -Non

homogeneous symmetric equations - Fredholm alternative – construction of greens

function for ordinary differential equations – using Green’s function in the solution

of boundary value.

Sections 17 to 20 of Chapter II of the prescribed text book II.


1. Describe homogeneous of integral equations with degenate kernel

2. Understanding the concept of Fredholmalternative,construction of greens

function for ODE.

3.Solving applications of Green’s function in the solution of boundary value.

Unit V

Learning

Out Comes

Calculus of variance: Introduction some typical problems of the subject – Euler

differential equation for an extremal – Isopermetric Problems.

Sections 47 to 49 of Chapter 9 of the prescribed text book I.


1. Solving problems of the Calculus of variance.

2. Determine concept of Euler differential equation for an extremal

3. Solving Isopermetric Problems.

Prescribed

Text Book

Reference

Books

Online

Source

1. Differential equation with Application historical notes by G.F.Simmons.

2.Integral Equation by Krasanov.

1. Sneddon I.N.,The Use of Integral Transforms, Tata McGraw Hill (1985).

2. GelfandI.M. andFominS.V., Calculus of Variations, Prentice Hall (1963).

3. Kenwal Ram P., Linear Integral Equations: Theory and Techniques, Academic

Press (1971).

1. https://www.researchgate.net/publication/267866066_Mathematical_Methods/download

2. https://physics.bgu.ac.il/~gedalin/Teaching/Mater/am.pdf.

3. https://www.elsevier.com/books/mathematical-methods/korevaar/978-1-4832-2813-6

Course Out

Comes

At the end of the course, students will

1. Laplace Transformation to solve initial and boundary value problems.;

2. To learn Fourier transformation and Z transformation and their applications to

relevant problems.;

3. To understand Hankel's Transformation to solve boundary value problem.;

4. Find solutions of linear integral equations of first and second type (Volterra and

Fredhlom)

5. Understand theory of calculus of variations to solve initial and boundary value

problems.

67

Course Code &

Course Name M 404(1)

Lattice Theory

Objectives

1. Lattice theory is the study of sets of objects known as lattices.

2. It is an outgrowth of the study of Boolean algebras.

3. Provides a framework for unifying the study of classes or ordered sets in mathematics.

4. Mathematical Logic, Boolean Algebra and its Applications, Switching circuit & Logic Gates,

Graphs and Trees.

5. Important theorems with constructive proofs, real life problems & graph theoretic algorithms.

SYLLABUS

Unit I

Learning

Out Comes

Partially Ordered sets- Diagrams- Special subsets of a poset -length- lower and

upper bounds- the minimum and maximum condition- the Jordan Dedekind chain

conditions -

Dimention functions.

Chapter I(sections 1 to 9) of the prescribed text book.


1. Distinguish Partially Ordered sets with diagrams

2. Classify the minimum and maximum conditions.

3. Learn applications Jordan Dedekind chain conditions.

Unit II

Learning

Out Comes

Algebras-lattices- the lattice theoretic duality principle- semilattices- lattices as

posets-diagrams of lattices- sub lattices, ideals

Chapter II(sections 10 to 16) of the prescribed text book


1.Classify Algebras-lattices- the lattice theoretic duality principle- semilattices

2. Determine the diagrams of lattices- sub lattices.

3. Understand the concept of ideals.

Unit III

Learning

Out Comes

Bound elements of Lattices-atoms and dualatomscomplements, relative

complements, semi complements-irreducible and prime elements of a lattice- the

homomorphism of a lattice-axioms systems of lattices.

Chapter II (sections 17 to 21) of the prescribed text book.


1. Determine the concept of atoms and dual atoms.

2. Understand the concept of compliments,semi compliments, relative

compliments.

3. Understand the concept of the homomorphism of a lattice-axioms systems of

lattices.

Unit IV

Boolean algebras, De Morgan formulae- Complete Boolean algebras- Boolean

algebras and Boolean rings- The algebra of relations- The lattice of propositions-

http://mathworld.wolfram.com/Lattice.html

http://mathworld.wolfram.com/BooleanAlgebra.html

68

Learning

Out Comes

Valuations of Boolean algebras.

Chapters VI(sections 42 to 47) of the prescribed text book .

1.Understand the concept of Boolean algebras, Demorgan formulae.

2.Determine the concept of Complete Boolean algebras- Boolean algebras and

Boolean rings-

3. Understand the concept of Valuations of Boolean algebras

Unit V

Learning

Out Comes

Ideals and dual ideals- Ideal chains- Ideal lattices- Distributive lattices and rings of

sets.

Chapter VIII(sections 53 to 55) of the prescribed text book .

1.Understand the concept of Ideals and dual ideals.

2.Determine the concept of Ideal chains- Ideal lattices

3. Understand the concept of Distributive lattices and rings of sets.

Prescribed

Text Book

Reference

Books

Online

Source

Introduction to Lattice Theory by Gabor Szasz, Academic Press, New York.

General Lattice Theory by G. Gratzer, Academic Press, New York.

1. http://www.math.hawaii.edu/~jb/lat1-6.pdf

2. http://www.math.ucla.edu/~yy26/works/Lattice%20Talk.pdf

3. http://boole.stanford.edu/cs353/handouts/book1.pdf

Course

Out Comes


1. Understand the concepts of Partially Ordered sets- Diagrams- Special subsets of

a poset -length- lower and upper bounds.

2 Understand the concepts of graph theory, Lattices, and Boolean Algebra in

analysis of various computer science applications.

3. Apply the knowledge of Boolean algebra in computer science for its wide

applicability in switching theory, building basic electronic circuits and design of

digital computers

4. Demonstrate knowledge and understandingthe concept of Complete Boolean

algebras- Boolean algebras and Boolean rings.

5. Understand the concept of Ideals and dual ideals, Distributive lattices and rings

of sets.

http://www.math.hawaii.edu/~jb/lat1-6.pdf

http://www.math.ucla.edu/~yy26/works/Lattice%20Talk.pdf

http://boole.stanford.edu/cs353/handouts/book1.pdf

69

Course Code


Theory Of Computations

Objectives: The goal of this course is to provide students with an understanding of basic

concepts in the theory of computation. At the end of this course students will:

1. Be able to construct finite state machines and the equivalent regular expressions.

2. Be able to prove the equivalence of languages described by finite state machines and regular

expressions.

3. Be able to construct pushdown automata and the equivalent context free grammars.

4. Be able to prove the equivalence of languages described by pushdown automata and context

free grammars.

5. Be able to prove the equivalence of languages described by Turing machines and Post

machines

SYLLABUS

Unit I

Learning

Out Comes

Sets, Relations, Special types of binary relations, logic preliminaries, finite-infinite

sets, fundamental proof techniques, alphabets, languages and their representations.


1. Write the Sets, Relations, Special types of binary relations .

2. Understand the concept of logic preliminaries, finite-infinite sets, fundamental

proof techniques.

3. Application of alphabets, languages and their representations.

Unit II

Learning

Out Comes

Deterministic finite automata, their equivalence, properties of languages accepted

by finite automata


1. Write the Deterministic finite automata.

2. Understand the concept of finite automata, their equivalence, properties.

3. Application of properties of languages accepted by finite automata.

Unit III

Learning

Out Comes

Regular expressions, (non) regular languages, Context free grammars.


1. Write the Regular expressions.

2. Understand the concept of regular languages.

3. Application of Context free grammars.

Unit IV

Learning

Out Comes

Context free languages, properties, push down automata, determinism and parsing.


1. Write the Context free languages .

2. Understand the concept of push down automata.

3. Application of determinism and parsing.

Unit V Turing machine, computing with Turing machines, combining Turing machines,

70

Learning

Out Comes

Extensions of Turing machines, nondeterministic Turing machines.


1. Write the Turing machine, computing with Turing machines.

2. Understand the concept ofcombining Turing machines.

3. Application ofExtensions of Turing machines, nondeterministic Turing

machines.

Prescribed

Text Book

Reference

Books

Online

Source

Hopcroft J. and Ullman J.D., Introduction to Automata Theory, Languages and

Computation.

1. Peter Linz, An Introduction to Formal Languages and Automata, Third

Edition, Jones and Bartlett, 2001.

2. Papadimitriou, Elements of the Theory of Computation, Prentice-Hall, 1998

3. John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman, "Introduction to

Automata Theory,Languages, and Computation", 2nd Edition, Prentice-Hall, 2001

4. Peter Dehning, Jack B. Dennis, “Machines, Languages and Computation”,

Second Edition, Prentice-Hall, 1978.

Course

Out Comes

1.Students will learn about a variety of issues in the mathematical development of

computer science theory, particularly finite representations for languages and

machines, as well as gain a more formal understanding of algorithms and

procedures.

2. In order to improve the pedagogy of this course, interactive animations of the

various automata using available simulators are recommended.

3. Acquire a full understanding and mentality of Automata Theory as the basis of

all computer science languages design

4. Have a clear understanding of the Automata theory concepts such as RE's,

DFA's, NFA's, Stack's, Turing machines, and Grammars.

5. Be able to minimize FA's and Grammars of Context Free Languages.

71

Course Code


Commutative Algebra II

Objectives

1. The course develops the theory of commutative rings.

2.These rings are of fundamental significance since geometric

3. Numbertheoretic ideas is described algebraically by commutative rings.

4. Knows basic definitions concerning elements in rings, classes of rings, and ideals in

commutative rings.

5. Can use algebraic tools which are important for many problems and much theory development

in algebra

SYLLABUS

Unit I

Learning

Out Comes

Integral dependence, the going-up theorem-Integrally closed integral domains.


1. Understand the concept of Integral dependence,

2. Write the going-up theorem

3. Understand the concept Integrally closed integral domains.

Unit II

Learning

Out Comes

The going-down theorem, valuation rings.


1. Write the going –down theorem.

2. Understand the concept of valuation rings.

3. Application of valuation rings.

Unit III

Learning

Out Comes

Chain Conditions.


1. Understand the concept of Chain conditions.

2. Write Ascending chain condition.

3. Write Descending chain condition.

Unit IV

Learning

Out Comes

Noetherian rings- Primary decomposition of Noetherian rings, Artin rings.


1. Understand the concept ofNoetherian rings.

2. Understand the concept of Primary decomposition of N.R.

3. Understand the concept ofArtin rings.

Unit V

Learning

Out Comes

Discrete valuation rings, Dedekind domains, Fractional ideals.


1. Express the concept of Discrete valuation rings.

2. Understand the concept of Dedekind domains.

3. Express the concept of Fractional ideals.

Prescribed

Text Book

1.Introduction to commutative algebra by M.F.Atiya and I.G.

Macdonald, Addison-Welsey Publishing Company, London.

72

Reference

Books

Online

Source

1. Basic Commutative Algebra by Balwant Singh, World scientific Publishing Co.

Pte. Ltd.

2. Commutative Algebra by N.S.Gopal Krishna, Second Edition.

1. web.mit.edu/18.705/www/13Ed.pdf

2. math.uga.edu/~pete/integral.pdf

3.https://www.jmilne.org/math/xnotes/CA.pdf

4. www.math.toronto.edu/jcarlson/A--M.pdf

Course

Out Comes

On Completion of this module, the learner will be able to

1. Knows basic definitions concerning elements in rings, classes of rings, and

ideals in commutative rings.

2. Know constructions like tensor product and localization, and the basic theory for

this.

3. Know basic theory for noetherian rings and Hilbert basis theorem.

4. Know basic theory for integral dependence, and the Noether normalization

lemma.

5. Have insight in the correspondence between ideals in polynomial rings, and the

corresponding geometric objects: affine varieties.





73

Course Code


Theory Of Linear Operators

Objectives: 1. Brief review of basic fact and terminology related to complete normed spaces and

linear functionals.

2. Finite, infinite and block matrices as linear operators. Schur test for boundedness. The adjoint

and the hermitianadjoint of an operator.

3. Finite rank and compact linear operators. Examples of integral operators. Operators with

compact resolvent. Application to some differential equations.

4. Hilbert-Schmidt operators. Examples of integral operators.

5. Spectral representation of compact selfadjoint operators in Hilbert spaces.

SYLLABUS

Unit I

Learning

Out Comes

Spectral Theory in Finite Dimensional Normed Spaces, Basic Concepts, Spectral

Properties of Bounded Linear Operators, Further Properties of Resolvent and

Spectrum, Use of Complex Analysis in Spectral Theory.


1. Express the concept of Spectral Theory in Finite Dimensional Normed Spaces

2. Understand the concept of Basic Concepts, Spectral Properties of Bounded

Linear Operators .

3. Express the concept of Resolvent and Spectrum, Use of Complex Analysis in

Spectral Theory.

Unit II

Learning

Out Comes

Banach Algebras, Further Properties of Banach Algebras, Compact Linear

Operators on Normed Spaces and Their Spectrum, Compact Linear Operators on

Normed Spaces, Further Properties of Compact Linear Operators, Spectral

Properties of Compact Linear Operators on Normed Spaces.


1. Express the concept of banach Algebras.

2. Understand the concept of Compact Linear Operators on Normed Spaces and

Their Spectrum.

3. Express the concept of Compact Linear Operators on Normed Spaces.

Unit III

Learning

Out Comes

Further Spectral Properties of Compact Linear Operators, Operator Equations

Involving Compact Linear Operators, Further Theorems of Fredholm

Type,Fredholm Alternative.


1. Express the concept of spectral Properties of Compact Linear Operators.

2. Understand the concept ofCompact Linear Operators.

3. Express the concept ofTheorems of Fredholm Type, Fredholm Alternative.

Unit IV

Spectral Theory of Bounded Self-Adjoint Linear Operators, Spectral Properties of

Bounded Self-Adjoint Linear Operators, Further Spectral Properties of Bounded

Self-Adjoint Linear Operators, Positive Operators, Square Roots of a Positive

Operator, Projection Operators, Further Properties of Projections

74

Learning

Out Comes


1. Express the concept of Spectral Theory of Bounded Self-Adjoint Linear

Operators.

2. Understand the concept ofSpectral Properties of Bounded Self-Adjoint Linear

Operators.

3. Express the concept ofPositive Operators, Square Roots of a Positive Operator.

Unit V

Learning

Out Comes

Spectral Family, Spectral Family of a Bounded Self-Adjoint Linear Operator,

Spectral Representation of Bounded Self-Adjoint Linear Operators, Extension of

the Spectral Theorem to Continuous Functions,Properties of the Spectral Family of

a Bounded Self- Adjoint Linear Operator.


1. Express the concept of Spectral Family of a Bounded Self-Adjoint Linear

Operator.

2. Understand the concept ofSpectral Representation of Bounded Self-Adjoint

Linear Operators.

3. Express the concept ofProperties of the Spectral Family of a Bounded Self-

Adjoint Linear Operator.

Prescribed

Text Book

Reference

Books

Online

Source

E. Kreyszig, Introductory Functional Analysis with Applications, JohnWiley&

Sons, New York, 1978.

1. P.R. Halmos, Introduction to Hilbert Space and the Theory of Spectral

Multiplicity, Second-Edition, Chelsea Publishing Co., New York, 1957.

2. N. Dunford and J.T. Schwartz, Linear Operators -3 Parts, Interscience/Wiley,

New York, 1958-71.

3. G. Bachman and L. Narici, Functional Analysis, Academic Press, York, 1966.

1.http://www-personal.acfr.usyd.edu.au/spns/cdm/resources/Kreyszig%20-

%20Introductory%20Functional%20Analysis%20with%20Applications.pdf

2.

Course

Out Comes

1. On successful completion of the course, students can opt for courses like

Operator Theory, Spectral Theory, Representation Theory etc.

2. Applications of spectral Thorem for compact operators. Polar decomposition.

3. Examples of spectral measures and applications of the Spectral Theorem.

4. Elementary properties of closed unbounded operators and solved applications of

differential operators. Cayley transform of symmetric oper

5. Spectral theorem for unbounded operators and Spectra of functions of operators.

http://www-personal.acfr.usyd.edu.au/spns/cdm/resources/Kreyszig%20-%20Introductory%20Functional%20Analysis%20with%20Applications.pdf

http://www-personal.acfr.usyd.edu.au/spns/cdm/resources/Kreyszig%20-%20Introductory%20Functional%20Analysis%20with%20Applications.pdf

75

Course Code


Wavelet Analysis

Objectives

1. The objective of this course is to establish the theory necessary to understand and use wavelets

and related constructions.

2. A particular emphasis will be put on constructions that are amenable to efficient algorithms,

since ultimately these are the ones that are likely to have an impact.

3. Study applications in signal processing, communications, and sensing where time-frequency

transforms like wavelets play an important role.

4. A particular emphasis will be put on constructions that are amenable to efficient algorithms,

since ultimately these are the ones that are likely to have an impact.

5. Study applications in signal processing, communications, and sensing where time-frequency

transforms like wavelets play an important role.

SYLLABUS

Unit I

Learning

Out Comes

An Overview: From Fourier analysis to wavelet analysis, The integral wavelet

transform and time-frequency analysis, Inversion formulas and duals,

Classification of wavelets, Multiresolution analysis, splines, and wavelets, Wavelet

decompositions and reconstructions.


1. Understand the concepts Fourier, Time-frequency window of wavelets, Discrete

wavelet transform, Haar wavelet and its Fourier transform.

2. Ability to distinguish Fourier and wavelet transforms.

3. Ability to write simple for some results on wavelet transforms.

Unit II

Learning

Out Comes

Fourier Analysis:Fourier and inverse Fourier transforms, Continuous-time

convolution and the delta function,Fourier transform of square-

integrablefunctions,Fourierseries,Basic convergence theory and Poisson's

summation formula


1. Understand Fourier and inverse Fourier transforms, continuous-time

convolution and the delta function.

2. Learn Fourier transform of square-integrable functions, Fourier series.

3. Ability to write proofs for some properties of wavelets.

Unit III

Learning

Out Comes

Wavelet Transforms and Time-Frequency Analysis:The Gabor transform,Short-time

Fourier transforms and the Uncertainty Principle,The integral wavelet

transform,Dyadic wavelets and inversions,Frames, Wavelet series


1. Understand the concepts of The Gabor transform, Short-time Fourier transforms

and the Uncertainty Principle.

2. Prepare Decomposition and reconstruction algorithm.


Unit IV

Cardinal Spline Analysis: Cardinal spline spaces, B-splines and their basic properties,

The two-scale relation and an interpolatory

76

Learning

Out Comes

graphical display algorithm,B-net representations and computation of cardinal

splines,Construction of spline approximation formulas, Construction of spline

interpolation formulas.


1. Learn Orthonormality in frequency domain, Numerical evaluation of scaling

function and wavelets.

2. Prepare Construction of spline approximation formulas, Construction of spline

interpolation formulas.


Unit V

Learning

Out Comes

Scaling Functions and Wavelets:Multiresolutionanalysis,Scaling functions with

finite two-scale relations, Direct-sum decompositions of L2(IR),Wavelets and their

duals,Linear –phase filtering, Compactly supported wavelets.


1. Understand the concepts of Multiresolution analysis,Scaling functions with

finite two-scale relations.

2. Prepare Decomposition and reconstruction algorithm.


Prescribed

Text Book

Reference

Text Books

Online

Source

1. C.K. Chui, An Introduction to Wavelets, Academic Press, New York, 1992.

2. I. Daubechies, Ten Lectures on Wavelets, CBS-NSF Regional Conferences in

Applied Mathematics, SIAM, Philadelphia, 1992.

3. O. Christensen, An Introduction to Frames and Riesz bases, Birkh• auser,

Boston, 2003.

1. Y. Meyer, Wavelets: Algorithms and Applications, SIAM, Philadelphia, 1993.

2. L. Debnath, Wavelet Transforms and their Applications, Birkh• auser, Boston,

2002.

3. M.W. Frazier, An Introduction to Wavelets through Linear Algebra, Springer,

New York, 1999.

4. M.K. Ahmad, Lecture Notes on Wavelet Analysis, Seminar Library, Department

of Maths, AMU, 2015.

5. A First course on Waveletsby Eugenio Hernandez and weiss, Ist edition CRC

press

1. http://math.bu.edu/people/mkon/Wavelets.pdf

2. http://web.iitd.ac.in/~sumeet/WaveletTutorial.pdf

3. https://inside.mines.edu/~whereman/talks/UIA-00-Wavelet-Lectures.pdf

Course

Out Comes

Upon completion of this course, you should be able to

1. Understand the terminology that is used in the wavelets literature.

2. Explain the concepts, theory, and algorithms behind wavelets from an

interdisciplinary perspective that unifies harmonic analysis (mathematics), filter

banks (signal processing), and multiresolution analysis (computer vision).

3. Master the modern signal processing tools using signal spaces, bases, operators

and series expansions.

4. Apply wavelets, filter banks, and multiresolution techniques to a problem at

hand, and justify why wavelets provide the right tool.

5. Research, present, and report a selected project within a specified time.

http://math.bu.edu/people/mkon/Wavelets.pdf

http://web.iitd.ac.in/~sumeet/WaveletTutorial.pdf

https://inside.mines.edu/~whereman/talks/UIA-00-Wavelet-Lectures.pdf

77

Course Code


PROGRAMMING IN C

Objectives

1.The course is designed to provide complete knowledge of C language.

2. Students will be able to develop logics which will help them to create programs, applications in

C.

3. Also by learning the basic programming constructs they can easily switch over to any other

language in future.

4. The course is oriented to those who want to advance structured and procedural programming

understating and to improve C programming skills.

5. The major objective is to provide students with understanding of code organization and

functional hierarchical decomposition with using complex data types.

SYLLABUS

Unit I

Learning

Out Comes

Definition of Algorithms x- Flow Charts writing Algorithms – some simple

examples to illustrate these concepts like finding the sum, GCD of two numbers –

swapping two variables, simple interest, area of a circle given its radius, area of a

triangle given all its sides, Largest of given three numbers, sowing a given

quadratic equation, sum of first ‘n’ natural numbers, Generation of Fibonacci

sequence - Given integer is prime or not.

(First four units of the prescribed text book I)


1.Illustrate the flowchart and design and algorithm for a given problem and to

develop IC programs using operators.

2. Ability to understand the concepts like finding the sum, GCD of two numbers –

swapping two variables, simple interest and etc.

3. Ability to write flow charts and algorithms for finding the sum, GCD of two

numbers – swapping two variables, simple interest and etc.

Unit II

Learning

Out Comes

Constants, Variables, and Data Types: Introduction - character set –C tokens –

keywords and identifiers constant – variables – Data types – Declaration of

variables – Assigning values to variables

Operators and Expression: Introduction - Arithmetic operators – Relational

operators – logical operators – Assignment operators - increment and decrement

operators – conditional operators – bitwise operators – special operators

Lab: writing C programmes which are related to problems on mathematics

(Unit 2 to Unit 3.9 of the prescribed text book II)


1. Understand the preliminaries for C language.

2. Ability to write the simple C programs for these concepts.

3. Ability to write C programs which are related to problems on mathematics

Unit III

Arithmetic expressions, evaluation of expressions, precedence of arithmetic

operators, some computational problems, type conversions in expressions, operator

precedence and associativity, Mathematical functions.Managing input and output

operations: Reading a character – writing a character formatted input – output


78

Learning

Out Comes

(Unit 3.10 to Unit 4 of the prescribed text book II)


1. Understand the mathematical preliminaries for C language.



Unit IV

Learning

Out Comes

Decision making and branching: Decision making with if statement – simple if

statement – the if else statement - nesting of if –else statement – The else of ladder

– the switch statement – the? Operators – the GO TO Decision Making and

Looping: Introduction - the while statement – the Do statement – the FOR

statement – jumps in loops


(Unit 5 to Unit 6 of the prescribed text book II)


1. Understand the concepts decision making statements, loops and nested loops.



Unit V

Learning

Out Comes

Arrays: Introduction - One dimensional arrays – two dimensional arrays –

initializing two- dimensional arrays – multidimensional arrays. Introduction to

pointers.

Lab: writing C programmes which are related to problems on mathematics.

(Unit 7 of the prescribed text book II)


1. Understand the declaration and implementation of arrays, pointers, functions

and structures.

2. Inscribe C programs using pointers and to allocate memory using dynamic

memory management functions.

3. Ability to write C programs which are related to problems on mathematics using

array and pointers

Prescribed

Text Book

Reference

Books

Online

Source

1. Programming techniques through ‘c’ by M.G. VenkateshMutry

2. C programming in Ans1 ‘C’ by E Balaguruswamy (Unit 2,3,4,) (second edition)

1. C Programming Absolute Beginner’s Guide (3rd Edition)’ by Greg Perry and

Dean Miller

2. The C Programming Language’ by Brian W. Kernighan and Dennis M. Ritchie.

3. C Programming: A Modern Approach (2nd Edition)’ by K. N. King

1. http://www.vssut.ac.in/lecture_notes/lecture1424354156.pdf

2. http://www.vssut.ac.in/lecture_notes/lecture1422486950.pdf

3. http://www.kciti.edu/wp-content/uploads/2017/07/cprogramming_tutorial.pdf

Course

Out Comes

After course completion the students will be able to

1. Develop conditional and iterative statements to write C programs.

2. Exercise user defined functions, data types, including structures and unions to

solve real time problems.

3. Exercise files concept to show input and output of files in C

4. To understand the file operations, character I/O, String I/O, file pointers and

importance of pre-processor directives.

5. Understand the declaration and implementation of arrays, pointers, functions

and structures and ability to write C programs which are related to problems on

mathematics using array and pointers.

http://www.vssut.ac.in/lecture_notes/lecture1424354156.pdf

http://www.vssut.ac.in/lecture_notes/lecture1422486950.pdf

http://www.kciti.edu/wp-content/uploads/2017/07/cprogramming_tutorial.pdf

79

Course Code


Semi Groups

• Objectives :To introduce the Concepts of Semigroups, monogenic Semigroups, Free

Semigroups, Ideals, Regular Semigroups, Simple and Q-Simple Semigroups, and their related

theories to develop working knowledge on these concepts and moreover

• 1. This course aims to expose the students to more liberal and powerful tools of Algebra that are

applicable in the present-day life.

• 2.To familiarise students with the elementary notions of semigroup theory.

• 3.To illustrate abstract ideas by applying them to a range of concrete examples of semigroups.

• 4.To study Green's relations and how these may be used to develop structure theorems for

semigroups.

• 5.To develop problem solving skills and to acquire knowledge on basic concepts of Semi groups,

Ideals and Rees’ congruencesand the structure of D-classes.

SYLLABUS

Unit I

Learning

Out Comes

Basic definition, monogenic semigroups, ordered sets, semilatttices and lattices,

binary relations, equivalences and congruences. Free semigroups, Ideals and Rees’

congruences.( Sections 1 to 4 of Chapter- I).


1.Understand basic definitions of Semigroups, Semilattices and Lattices, and their

basic Results.

2.student able to appreciate the importance of semigroup theory in abstract

algebra;

3.Familiar with the most important classes of semigroups and have an

understanding of the structure of important examples, such as the most famous

transformation semigroups.

Unit II

Learning

Out Comes

Lattices of equivalences and congruences, Green’s equivalences, the structure of

D-classes, regular semigroups- Simple and Q- simple semigroups. . ( Sections 5 to

8 of Ch. I ).

Upon completion of this unit, the student will be able to:

1.Understandcongrancesand Green’s equivalences, and also find Structure of D-

Classes.

2.The basic ideas of the subject, including Green’s relations, and be able to handle

the algebra of semigroups in a comfortable way.

3.The role of structure theorems, and be able to use Rees' theorem for completely

Q-simple semigroups.

80

Unit III

Learning

Out Comes

Principal factors, Rees’ theorem, Primitive idempotents.Congruences on

completely Q-simple semi groups.(Sections 1 to 3 of Chapter III ).


1. Analyze Simple and Q-Simple Semigroups, and Rees’s Theorem.

2.Demonstrate knowledge of the primitive idempotents;

3.Write precise and accurate mathematical definitions of objects in Semi groups;

Unit IV

Learning

Out Comes

The lattice of congruences on a completely 0-simple semigroup, Finite congruence

free semigroups.(Sections 4 to 6 of Chapter III ).


1. Describe Congruences on Completely O-Simple Semigroups and Finite

Congruences.

2.Identify and develop free semigroup models from the verbal description of the

real system.

3. Understand the definitions of (completely) (0)-simple semigroups and the proofs

of some of the main theorems in this section.

Unit V

Learning

Out Comes

Union of Groups, Semi lattices of groups, bands, free bands, varieties of bands.

(Sections 1 to 5 of Chapter IV ).


1. Students will demonstrate knowledge and comprehension of basic principles of

semi-lattice of groups.

2. Students will be able to apply basic principles of bands and varieties of bands in

simple mathematical problem solving involving mathematical structures.

3.Use mathematical definitions to identify and construct examples and to

distinguish examples from non-examples;

Prescribed

Text Book

Reference

Books

Online

Source

An introduction to semi group theory by J.M. Howie, 1976, Academic press,

New York.

1.John M. Howie: Fundamentals of semigroup theory, Clarendon press, Oxford,

1995.

2. A. H. Clifford and G. B. Preston: The Algebraic theory of semi groups, Vol. 1,

and 2, Mathematical surveys of the AMS, 1961 and 1967.

3. P. M. Higgins: Techniques of Semi Group Theory, Oxford University Press,

1992.

1. https://pdfs.semanticscholar.org/3193/9dfde70be855c8919462216c0801b5d4a8de.pdf

2. https://www.ams.org/books/surv/007.1/surv007.1-endmatter.pdf

Course

Out Comes Having successfully completed this course student will be able to:

1. the basic properties of Green's relations and use these in an appropriate way

2. Construct new semigroups using congruences.

3.To appreciate the importance of semigroup theory in abstract algebra.

4.To learn and feel that learnig further advance tools of this discipline will equip

them to apply these tools to the huge world of Automata, Languages and

Machines.

5. The student realizes the richness of properties enjoyed by Semigroups, an

https://pdfs.semanticscholar.org/3193/9dfde70be855c8919462216c0801b5d4a8de.pdf

81

algebraic structure with fewer facilities than Groups.

Course Code


Financial Mathematics

Objectives

1. This is an introductory course in Financial Mathematics.

2. Student will learn about the different types of interest (simple interest, discount interest,

compound interest), annuities, debt retirement methods, investing in stocks and bonds. Time

permitting, more advanced topics will also be covered.

3. Apply logical thinking to problem solving in context.

4. Use appropriate technology to aid problem solving.

5. Define interest rate risk in terms of duration and convexity of fixed interest products.

SYLLABUS

Unit I

Learning

Out Comes

The Measurement of Interest: Introduction, The accumulation and amount

functions, The effective rate of interest, Simple interest, Compound interest,

Present value, The effective rate of discount, Nominal rates of interest and

discount, Forces of interest and discount, Varying interest, Summary of results.

Solution of Problems in Interest: Introduction, The basic problem, Equation of

value, Unknown time, Unknown rate of interest, Determining time periods,

Practical examples.


1. Define and recognize the definitions of the following terms: interest rate (rate of

interest), simple interest, compound interest, accumulation function, future value,

current value, present value, etc.

2. Give any one of the effective interest rate, the nominal interest rate.

3. Understand the concepts.

Unit II

Learning

Out Comes

Basic Annuities: Introduction, Annuity-immediate, Annuity-due, Annuity values

on any date, Perpetuities, Unknown time, Unknown rate of interest, Varying

interest, Annuities not involving compound interest.


1. Understand the concepts Unknown rate of interest, Varying interest.

2.Write the equation of value given a set of cash flows and an interest rate.

3.Apply these concepts in real world problems.

Unit III

Learning

Out Comes

More General Annuities: Introduction, Differing payment and interest conversion

periods, Annuities payable less frequently than interest convertible, Annuities

payable more frequently than interest convertible, Continuous annuities, Payments

varying in arithmetic progression, Payments varying in geometric progression,

More general varying annuities, Continuous varying annuities, Summary of

results.


1. Understand the concepts carefully.

2.Describe in detail the various types of annuities and perpetuities and use them to

solve financial transaction problems.


Unit IV Amortization Schedules and Sinking Funds: Introduction, Finding the outstanding

82

Learning

Out Comes

loan balance, Amortization schedules, Sinking funds, Differing payment periods

and interest conversion periods, Varying series of payments, Amortization with

continuous payments, Step-rate amounts of principal.


1.Understand the concepts

2.Explain the details of arbitrage and its use in the valuation of forward contracts.

Employ term structure of interest rates to calculate forward and spot rates.


Unit V

Learning

Out Comes

Bonds and Other Securities: Introduction, Types of securities, Price of a bond,

Premium and discount, Valuation between coupon payment dates, Determination

of yields rates, Callable and putable bonds, Serial bonds, some generalizations,

other securities, Valuation of securities.

Yield Rates: Introduction, Discounted cash flow analysis, Uniqueness of the yield

rate, Reinvestment rates, Interest measurement of a fund, Time-weighted rates of

interest, Portfolio methods and investment year methods, Short sales, Capital

budgetingbasic technique and other technique.


1.Define interest rate risk in terms of duration and convexity of fixed interest

products.

2. Find Employ term structure of interest rates to calculate forward and spot rates.

3. Apply these methods in real life.

Prescribed

Text Book

Reference

Books

Online

Source

1. Stephen G. Kellison, The Theory of Interest, 3rd Edition. McGraw Hill

International Edition (2009).

2. R. J. Elliott and P. E. Kopp, Mathematics of Financial Markets, Springer (1999)

Harshbarger, R.J. & Reynolds, J.J., Mathematical Applications for the

Management, Life and Social Sciences 12th ed.

1. http://www1.maths.leeds.ac.uk/~jitse/math1510/notes-all.pdf

2. https://people.kth.se/~lang/finansmatte/fin_note.pdf

Course

Out Comes

At the end of the course students will be expected to

1. Determine and select the most appropriate standard mathematical, statistical and

computing methods appropriate for specifying mathematical problems in banks

and other financial institutions through a critical understanding of the relative

advantages of these methods, and to develop extensions to these methods

appropriate for the solution of non-standard problems;

2. Know the main features of models commonly applied in financial firms, be able

to express these mathematically and be able to appraise their utility and

effectiveness;

3. Explain and critically appraise the rationale for the selection of mathematical

tools used in the analysis of common financial problems;

4. Be able to demonstrate the appropriateness of modelling or numerical solutions

in analysing common problems in banks and other financial institutions;

5. Be able to select and apply numerical solutions in some areas of finance.

http://www1.maths.leeds.ac.uk/~jitse/math1510/notes-all.pdf

https://people.kth.se/~lang/finansmatte/fin_note.pdf